English words of Greek origin
The Greek language has contributed to the English vocabulary in five main ways:
 vernacular borrowings, transmitted orally through Vulgar Latin directly into Old English, e.g., 'butter' (Old English butere from Latin butyrum < βούτυρον), or through French, e.g., 'ochre'.
 learned borrowings from classical Greek texts, often via Latin, e.g., 'physics' (< Latin physica < Greek τὰ φυσικά);
 a few borrowings transmitted through other languages, notably Arabic scientific and philosophical writing, e.g., 'alchemy' (< χημεία);
 coinages in post-classical Latin or modern languages using classical Greek roots, e.g., 'telephone' (< τῆλε + φωνή) or a mixture of Greek and other roots, e.g., 'television' (< Greek τῆλε + English 'vision' < Latin visio); these are often shared among the modern European languages, including Modern Greek;
 direct borrowings from Modern Greek, e.g., ouzo.

The post-classical coinages are by far the most numerous of these.

Indirect and direct borrowings

Since the living Greek and English languages were not in direct contact until modern times, borrowings were necessarily indirect, coming either through Latin (through texts or various vernaculars), or from Ancient Greek texts, not the living spoken language.

Vernacular borrowings via Romance languages

Some Greek words were borrowed into Latin and its descendants, the Romance languages.  English often received these words from French. Some have remained very close to the Greek original, e.g. lamp, Latin lampas, Greek λαμπάς. In others, the phonetic and orthographic form has changed considerably. For instance, place was borrowed both by Old English and by French from Latin platea, itself borrowed from Greek πλατεία (ὁδός) 'broad (street)'; the Italian piazza and Spanish plaza have the same origin, and have been borrowed into English in parallel.  The word olive comes through the Romance from the Latin word olīva, which in turn comes from the Greek  (elaíwā). A later Greek word,  (boútȳron) becomes Latin butyrum and eventually English 'butter'. A large group of early borrowings, again transmitted first through Latin, then through various vernaculars, comes from Christian vocabulary: chair << καθέδρα (cf. 'cathedra'), bishop << ἐπίσκοπος (epískopos 'overseer'), priest << πρεσβύτερος  (presbýteros 'elder'), and church < Old English cirice, circe < probably κυριακή [οἰκία] (kȳriakḗ [oikía] 'lord's [house]'). In some cases, the orthography of these words was later changed to reflect the Greek – and Latin – spelling: e.g., quire was respelled as choir in the 17th century. Sometimes this was done incorrectly: ache is from a Germanic root; the spelling ache reflects Samuel Johnson's incorrect etymology from Greek ἄχος.

Learned borrowings

Many more words were borrowed by scholars writing in Medieval and Renaissance Latin.  Some words were borrowed in essentially their original meaning, often transmitted through classical Latin: topic, type, physics, iambic, eta, necromancy, cosmopolite. A few result from scribal errors: encyclopedia < ἐγκύκλιος παιδεία 'the circle of learning', not a compound in Greek; acne (skin condition) < erroneous ἀκνή < ἀκμή 'high point, acme'. Some kept their Latin form, e.g., podium < πόδιον.

Others were borrowed unchanged as technical terms, but with specific, novel meanings: telescope < τηλεσκόπος 'far-seeing' refers to an optical instrument for seeing far away rather than a person who can see far into the distance; phlogiston < φλογιστόν 'burnt thing' is a supposed fire-making potential rather than something which has been burned, or can be burned; bacterium < βακτήριον 'stick (diminutive)' is a kind of microorganism rather than a small stick or staff. This also applies to combining forms used in neologisms: -cyte or cyto- < κύτος 'container' means biological cells, not arbitrary containers; -oma < -ωμα, a generic morpheme forming deverbal nouns such as diploma 'a folded thing' and glaucoma 'greyness', comes to have the very narrow meaning of 'tumor' or 'swelling', on the model of words like carcinoma < καρκίνωμα. For example, melanoma does not come from Greek μελάνωμα 'blackness', but rather from the modern combining forms melano- 'dark (in biology)' + -oma 'tumor'.

Usage in neologisms

But by far the largest Greek contribution to English vocabulary is the huge number of scientific, medical, and technical neologisms that have been coined by compounding Greek roots and affixes to produce novel words which never existed in the Greek language: utopia (1516, οὐ 'not' + τόπος 'place'), zoology (1669, ζῷον + λογία), hydrodynamics (1738, ὕδωρ + δυναμικός), photography (1834, φῶς + γραφικός), oocyte (1895, ᾠόν + κύτος), helicobacter (1989, ἕλιξ + βακτήριον).  So it is really the combining forms of Greek roots and affixes that are borrowed, not the words. Such terms are coined in all the European languages, and spread to the others freely—including to Modern Greek as reborrowings. Traditionally, these coinages were constructed using only Greek morphemes, e.g., metamathematics, but  increasingly, Greek, Latin, and other morphemes are combined, as in television (Greek τῆλε + Latin vision), metalinguistic (Greek μετά + Latin lingua + Greek -ιστής + Greek -ικος), and garbology (English garbage + Greek -ολογία).  These hybrid words were formerly considered to be 'barbarisms'.

Some derivations are idiosyncratic, for example, utopia and gas (< χάος) are irregular both in formation and in spelling; hadron < ἁδρός with the suffix -on, itself abstracted from anion ἀνιόν; henotheism < ἑνό(ς) 'one' + θεός 'god', though eno- is not used as a prefix in Greek; and taxonomy < τάξις 'order' + -nomy -νομία 'study of', where the "more etymological form" is taxinomy, as found in the word ταξίαρχος 'taxiarch' and the neologism taxidermy; Modern Greek uses ταξινομία in its reborrowing.

Many Greek affixes such as anti- and -ic have become productive in English, combining with arbitrary English words: antichoice, Fascistic. Some portmanteau words in English have led to suffixes based on Greek words, but which are not suffixes in Greek, e.g. -athon or -a-thon (walkathon, from walk and (mar)athon) and -nomics (Reaganomics, from Reagan and (eco)nomics).

Through other languages

Some Greek words were borrowed through Arabic and then Romance. Many are learnèd: alchemy (al- + χημεία or χημία), elixir (al- + ξήριον), alembic (al- + ἄμβιξ); but others are popular: botargo (ᾠοτάριχον), tajine (τάγηνον), and possibly quintal (κεντηνάριον < Latin centenarium (pondus)). Curiously, chemist appears to be a back-formation from alchemist.

A few words took other routes. Seine (a kind of fishing net) comes from a West Germanic form *sagīna, from Latin sagēna, from Greek σαγήνη. Effendi comes from Turkish, borrowed from medieval Greek αυθέντης /afˈθendis/ 'lord'.

Vernacular or learned doublets

Some Greek words have given rise to etymological doublets, being borrowed both through a later learned, direct route, and earlier through an organic, indirect route:

 ἀμυγδάλη amygdala, almond;
 ἀντίφωνα antiphon, anthem;
 ἀσϕόδελος asphodel, daffodil;
 αὐθεντικός authentic, effendi (αὐθέντης via Turkish);
 βάλσαμον balsam (probably itself a borrowing from Semitic), balm;
 βλάσφημος blasphemy, blame;
 βούτυρον butyr(ic), butter;
 διάβολος diabol(ic), devil;
 δραχμή drachma, dram, also dirhem via Arabic;
 ἔλαιον elaeo-, oil, olive, oleum;
 ἐλεημοσύνη eleemosynary, alms;
 ἐπίσκοπος episcop(al), bishop;
 ζῆλος zeal, jealous;
 καθέδρα cathedra(l), chair, chaise;
 κέρας/κέρατ- 'horn' keratin, carat via Arabic;

 κόλπος 'lap, womb, hollow, bay' colp(itis), gulf
 κυβερνᾶν cybernetics, govern
 πάπυρος papyrus, paper;
 πόδιον podium, pew;
 πρεσβύτερος presbyter, priest;
 πυξίς pyx(is), box;
 σκάνδαλον scandal, slander;
 τρίπους/τρίποδ- tripod, trivet;
 τύμπανον 'drum' tympanum 'eardrum', timbre, timpani;
 φρενετικός frenetic, frantic;
 χειρουργός chirurgical, surgeon;
 χορός chorus, choir;
 χρῖσμα chrism, cream;
 χρῑστιᾱνός Christian, christen, cretin?
 ὥρα horo(scope), hour.

Other doublets come from differentiation in the borrowing languages:

 γραμματική grammatic(al): grammar, glamor, grimoire;
 δίσκος discus: disc, dish, dais, and desk;
 κιθάρα cither: guitar, cithren, zither, gittern, cittern, etc.;

 κρύπτη crypt: grotto, (under)croft;
 παραβολή parabola: parable; additional doublets in Romance give palaver, parol, and parole;
 ϕαντασία phantasy: fantasy; fancy in 15th-century English.

From modern Greek

Finally, with the growth of tourism and emigration, some words reflecting modern Greek culture have been borrowed into English—many of them originally borrowings into Greek themselves: retsina, souvlaki, taverna (< Italian), ouzo (disputed etymology), moussaka (< Turkish < Arabic), baklava (< Turkish), feta (< Italian), bouzouki (< Turkish), gyro (the food, a calque of Turkish döner).

Greek as an intermediary

Many words from the Hebrew Bible were transmitted to the western languages through the Greek of the Septuagint, often without morphological regularization: rabbi (ραββί), seraphim (σεραφείμ, σεραφίμ), paradise (παράδεισος < Hebrew < Persian), pharaoh (Φαραώ < Hebrew < Egyptian).

Written form of Greek words in English

Many Greek words, especially those borrowed through the literary tradition, are recognizable as such from their spelling.  Latin had standard orthographies for Greek borrowings: Greek  was written as 'y',  as 'e',  as 'ch',  as 'ph',  as 'c', rough breathings as 'h', both  and  as 'i', etc. These conventions, which originally reflected pronunciation, have carried over into English and other languages with historical orthography, like French. They make it possible to recognize words of Greek origin, and give hints as to their pronunciation and inflection.

The romanization of some digraphs is rendered in various ways in English. The Ancient Greek diphthongs  and  may be spelled in three different ways in English: the Latinate digraphs ae and oe; the ligatures æ and œ; and the simple letter e. The ligatures have largely fallen out of use worldwide; the digraphs are uncommon in American usage, but remain common in British usage.  The spelling depends mostly on the variety of English, not on the particular word. Examples are: encyclopaedia  / encyclopædia / encyclopedia, haemoglobin / hæmoglobin / hemoglobin, oedema / œdema / edema. Some words are almost always written with the digraph or ligature: amoeba / amœba, rarely ameba; Oedipus / Œdipus rarely Edipus; others are almost always written with the single letter: sphære and hæresie were obsolete by 1700; phænomenon by 1800; phænotype and phænol by 1930. The verbal ending  is spelled -ize in American English and -ise or -ize in British English.

Since the 19th century, a few learned words were introduced using a direct transliteration of Ancient Greek and including the Greek endings, rather than the traditional Latin-based spelling: nous (νοῦς), koine (κοινή), hoi polloi (οἱ πολλοί), kudos (κύδος), moron (μωρόν), kubernetes (κυβερνήτης). For this reason, the Ancient Greek digraph  is rendered differently in different words—as i, following the standard Latin form: idol < εἴδωλον; or as ei, transliterating the Greek directly: eidetic (< εἰδητικός), deixis, seismic. Most plurals of words ending in -is are -es, using the regular Latin plural rather than the Greek -εις: crises, analyses, bases, with only a few didactic words having English plurals in -eis: poleis, necropoleis, and acropoleis (though acropolises is by far the most common English plural).

Most learned borrowings and coinages follow the Latin system, but there are some irregularities: eureka (cf. heuristic), kinetic (cf. cinematography), krypton (cf. cryptic), acolyte (< ἀκόλουθος; acoluth would be the etymological spelling, but acolythus, acolotus, acolithus are all found in Latin), stoichiometry (< στοιχεῖον, regular spelling would be st(o)echio-). Aneurysm was formerly often spelled aneurism on the assumption that it uses the usual -ism ending.

Some words whose spelling in French and Middle English did not reflect their Greco-Latin origins were refashioned with etymological spellings in the 16th and 17th centuries: caracter became character and quire became choir.

In some cases, a word's spelling clearly shows its Greek origin. If it includes ph or includes y between consonants, it is very likely Greek, with some exceptions, such as nephew, cipher, triumph. If it includes rrh, phth, or chth; or starts with hy-, ps-, pn-, or chr-; or the rarer pt-, ct-, chth-, rh-, x-, sth-, mn-, tm-, gn- or bd-, then it is Greek, with some exceptions: gnat, gnaw, gneiss.  Other exceptions: ptarmigan is from a Gaelic word, the p having been added by false etymology; style is probably written with a 'y' because the Greek word στῦλος 'column' and the Latin word stilus 'stake, pointed instrument' were confused. The word trophy, though ultimately of Greek origin, did not have a  but a  in its Greek form, τρόπαιον.

Pronunciation

In clusters such as ps-, pn-, and gn- which are not allowed by English phonotactics, the usual English pronunciation drops the first consonant (e.g., psychology) at the start of a word; compare gnostic [nɒstɪk] and agnostic [ægnɒstɪk]; there are a few exceptions: tmesis [t(ə)miːsɪs]. Initial x- is pronounced z.  Ch is pronounced like k rather than as in "church": e.g., character, chaos. The consecutive vowel letters 'ea' are generally pronounced separately rather than forming a single vowel sound when transcribing a Greek εα, which was not a digraph, but simply a sequence of two vowels with hiatus, as in genealogy or pancreas; cf., however, ocean ωκεανός; the 'ea' in zeal comes irregularly from the η in ζήλος.

Some sound sequences in English are only found in borrowings from Greek, notably initial sequences of two fricatives, as in sphere. Most initial /z/ sounds are found in Greek borrowings.

Inflectional endings and plurals

Though many English words derived from Greek through the literary route drop the inflectional endings (tripod, zoology, pentagon) or use Latin endings (papyrus, mausoleum), some preserve the Greek endings: -ον: phenomenon, criterion, neuron, lexicon; plasma (-∅; cf. protoplasm), drama, dilemma, trauma; -ος: chaos, ethos, asbestos, pathos, cosmos; -ς: climax (ξ x = k + s), helix, larynx, eros, pancreas, atlas; -η: catastrophe, agape, psyche; -ις: analysis, basis, crisis, emphasis; -ης: diabetes, herpes, isosceles. In cases like scene, zone, fame, though the Greek words ended in -η, the silent English e is not derived from it.

In the case of Greek endings, the plurals sometimes follow the Greek rules: phenomenon, phenomena; tetrahedron, tetrahedra;  crisis, crises; hypothesis, hypotheses; polis, poleis; stigma, stigmata; topos, topoi; cyclops, cyclopes; but often do not: colon, colons not *cola (except for the very rare technical term of rhetoric); pentathlon, pentathlons not *pentathla; demon, demons not *demones; climaxes, not *climaces. Usage is mixed in some cases: schema, schemas or schemata; lexicon, lexicons or lexica; helix, helixes or helices; sphinx, sphinges or sphinxes; clitoris, clitorises or clitorides.  And there are misleading cases: pentagon comes from Greek pentagonon, so its plural cannot be *pentaga; it is pentagons -- the Greek form would be *pentagona (cf. Plurals from Latin and Greek).

Verbs

A few dozen English verbs are derived from the corresponding Greek verbs; examples are baptize, blame and blaspheme, stigmatize, ostracize, and cauterize. In addition, the Greek verbal suffix -ize is productive in Latin, the Romance languages, and English: words like metabolize, though composed of a Greek root and a Greek suffix, are modern compounds. A few of these also existed in ancient Greek, such as crystallize, characterize, and democratize, but were probably coined independently in modern languages; this is particularly clear in cases like allegorize and synergize, where the Greek verbs ἀλληγορεῖν and συνεργεῖν do not end in -ize at all. Some English verbs with ultimate Greek etymologies, like pause and cycle, were formed as denominal verbs in English, even though there are corresponding Greek verbs, παῦειν/παυσ- and κυκλεῖν.

Borrowings and cognates

Greek and English share many Indo-European cognates. In some cases, the cognates can be confused with borrowings. For example, the English mouse is cognate with Greek μῦς /mys/ and Latin mūs, all from an Indo-European word *mūs; they are not borrowings. Similarly, acre is cognate to Latin ager and Greek αγρός, but not a borrowing; the prefix agro- is a borrowing from Greek, and the prefix agri- a borrowing from Latin.

Phrases

Many Latin phrases are used verbatim in English texts—et cetera (etc.),  ad nauseam, modus operandi (M.O.), ad hoc, in flagrante delicto, mea culpa, and so on—but this is rarer for Greek phrases or expressions: hoi polloi 'the many', eureka 'I have found (it)', kalos kagathos 'beautiful and virtuous', hapax legomenon 'once said', kyrie eleison 'Lord, have mercy'.

Calques and translations

Greek technical terminology was often calqued in Latin rather than borrowed, and then borrowed from Latin into English. Examples include (grammatical) case from Latin casus 'an event, something that has fallen', a semantic calque of Greek πτώσις 'a fall'; nominative, from Latin nōminātīvus, a translation of Greek ὀνομαστική; adverb, a morphological calque of Greek  ἐπίρρημα as ad- + verbum; magnanimus, from Greek μεγάθυμος, literally 'great spirit'. Essence is from Latin essentia, which was constructed from the notional present participle *essens, imitating Greek οὐσία. Substance is Latin substantia, a calque of Greek υπόστασις (cf. hypostasis). Cicero coined moral on analogy with Greek ηθικός. Recant is modeled on παλινῳδεῖν.<ref>A.J. Woodman, "O MATRE PVLCHRA: The Logical Iambist: To the memory of Niall Rudd", The Classical Quarterly 68:1:192-198 (May 2018) , footnote 26</ref>

Greek phrases were also calqued in Latin, then borrowed or translated into English. A commonplace is an English calque of the Latin locus communis, itself a calque of Greek κοινός τόπος. The Latin phrase deus ex machina (‘god out of the machine’) was calqued from the Greek  ἀπὸ μηχανῆς θεός (apò mēkhanês theós). Materia medica is a short form of Dioscorides' De Materia Medica, the Latin translation of Περὶ ὕλης ἰατρικῆς. Q.E.D. (quod erat demonstrandum) is a calque of ὅπερ ἔδει δεῖξαι. Subject matter is a calque of Latin subiecta māteria, itself a calque of Aristotle's phrase ἡ ὑποκειμένη ὕλη. Wisdom tooth came to English from Latin dentes sapientiae, from Arabic aḍrāsu 'lḥikmi, from Greek σωϕρονιστῆρες, used by Hippocrates. Political animal is from Greek πολιτικὸν ζῷον (in Aristotle's Politics). Quintessence is post-classical Latin quinta essentia, from Greek πέμπτη οὐσία.

The Greek word εὐαγγέλιον has come into English both in borrowed forms like evangelical and the form gospel, an English calque (Old English gód spel 'good tidings') of Latin bona adnuntiatio, itself a calque of the Greek.

Statistics
The contribution of Greek to the English vocabulary can be quantified in two ways, type and token frequencies: type frequency is the proportion of distinct words; token frequency is the proportion of words in actual texts.

Since most words of Greek origin are specialized technical and scientific coinages, the type frequency is considerably higher than the token frequency.  And the type frequency in a large word list will be larger than that in a small word list. In a typical English dictionary of 80,000 words, which corresponds very roughly to the vocabulary of an educated English speaker, about 5% of the words are borrowed from Greek.

Most common

Of the 500 most common words in English, 18 are of Greek origin: place (rank 115), problem (121), school (147), system (180), program (241), idea (252), story (307), base (328), center (335), period (383), history (386), type (390), music (393), political (395), policy (400), paper (426), phone (480), economic (494).