Tiberian Hebrew
Tiberian Hebrew is the canonical pronunciation of the Hebrew Bible (Tanakh) committed to writing by Masoretic scholars living in the Jewish community of Tiberias in ancient Galilee c. 750–950 CE under the Abbasid Caliphate. They wrote in the form of Tiberian vocalization, which employed diacritics added to the Hebrew letters: vowel signs and consonant diacritics (nequdot) and the so-called accents (two related systems of cantillation signs or te'amim). These together with the marginal notes masora magna and masora parva make up the Tiberian apparatus. Although the written vowels and accents came into use in around 750 CE, the oral tradition that they reflect is many centuries older, with ancient roots.
Tiberian Hebrew is the canonical pronunciation of the Hebrew Bible (Tanakh) committed to writing by Masoretic scholars living in the Jewish community of Tiberias in ancient Galilee c. 750–950 CE under the Abbasid Caliphate. They wrote in the form of Tiberian vocalization, which employed diacritics added to the Hebrew letters: vowel signs and consonant diacritics (nequdot) and the so-called accents (two related systems of cantillation signs or te'amim). These together with the marginal notes masora magna and masora parva make up the Tiberian apparatus.
Although the written vowels and accents came into use in around 750 CE, the oral tradition that they reflect is many centuries older, with ancient roots.

Sources
Today's Hebrew grammar books do not teach the Tiberian Hebrew that was described by the early grammarians. The prevailing view is that of David Qimchi's system of dividing the graphic signs into "short" and "long" vowels. The values assigned to the Tiberian vowel signs reveals a Sephardi tradition of pronunciation (the dual quality of qames (אָ) as /a/, /o/; the pronunciation of simple sheva (אְ) as /ɛ̆/).
The phonology of Tiberian Hebrew can be gleaned from the collation of various sources:

The Aleppo Codex of the Hebrew Bible and ancient manuscripts of the Tanakh cited in the margins of early codices, all of which preserve direct evidence in a graphic manner of the application of vocalization rules such as the widespread use of chateph vowels where one would expect simple sheva, thus clarifying the color of the vowel pronounced under certain circumstances. Most striking is the use of chateph chireq in five words under a consonant that follows a guttural vocalized with regular chireq (as described by Israel Yeivin) as well as the anomalous use of the raphe sign over letters that do not belong to בגדכפ"ת or א"ה.
The explicit statements found in grammars of the 10th and the 11th centuries, including the Sefer haQoloth ספר הקולות of Moshe ben Asher (published by N. Allony); the Sefer Dikdukei ha-Te'amim (ספר דקדוקי הטעמים Grammar or Analysis of the Accents) of Aaron ben Moses ben Asher; the anonymous works entitled Horayath haQoré הורית הקורא (G. Khan and Ilan Eldar attribute it to the Karaite Abu Alfaraj Harun); the Treatise on the Schwa (published by Kurt Levy from a Genizah fragment in 1936), and Ma'amar haschewa מאמר השוא (published from Genizah material by Allony); the works of medieval Sephardi grammarians including Abraham Ibn Ezra and Judah ben David Hayyuj. In the case of the latter two, it is evident that the chain of transmission is breaking down or that their interpretations are influenced by local tradition.
Ancient manuscripts that preserve similar dialects of Hebrew or Palestinian Aramaic but are vocalized in Tiberian signs in a "vulgar" manner and so reveal a phonetic spelling rather than a phonemic spelling. They include the so-called "pseudo-Ben Naphtali" or "Palestinian-Sephardi" vocalized manuscripts, which generally conform to the rules enumerated below, such as pronouncing sheva as /ĭ/ before consonantal yod, as in /bĭji/ בְּיִ.
Other traditions such as the vocalization of the Land of Israel and (to a lesser extent) the Babylonian vocalization. Each community (Palestinian, Tiberian, Babylonian) developed systems of notation for pronunciation in each dialect, some of which are common among the traditions.
The transcriptions of Biblical text into Arabic characters and then vocalized with Tiberian signs (by members of the Karaite community) provide an aid to pronouncing Tiberian Hebrew, especially for syllable structure and vowel length (which is marked in Arabic by matres lectionis and the sign sukun).
Various oral traditions, especially those of Yemenite Hebrew pronunciation and the Karaite tradition, have both preserved old features that correspond to Tiberian tradition, such as the pronunciation of schwa according to its proximity to gutturals or yod.

Phonology
Consonants
Tiberian Hebrew has 29 consonantal phonemes, represented by 22 letters. The sin dot distinguishes between the two values of ש‎, with a dot on the left (שׂ‎) being pronounced the same as the letter Samekh. The letters בגדכפת‎ (begadkefat) had two values each: plosive and fricative.

The following are the most salient characteristics of the Tiberian Hebrew consonantal pronunciation:

Waw ו conjunctive was read, before the labial vowels (בומ״ף) and shva (אְ), as אוּ /ʔu/, rather than וֻ /wu/ (as is the case in some eastern reading traditions).
The threefold pronunciation of Resh ר. Even though there is no agreement as to how it was pronounced, the rules of distribution of such pronunciation is given in הורית הקורא Horayath haQoré:
"Normal" Resh /ʀ/ pronounced thus (according to Eldar, as a uvular sound [ʀ]) in all other instances (except for the circumstances described below): אוֹר [ʔoʀ]
The "peculiar" resh [r] before or after Lamed or Nun, any of the three being vocalized with simple sheva and Resh after Zayin ז, Daleth ד, Samekh ס, Sin שׂ, Taw ת, Tzadi צ, Teth ט, any of them punctuated with simple sheva: יִשְׂרָאֵל [jisrɔˈʔel], עָרְלָה [ʕɔrˈlɔ]. Because of the proximity of a dental consonant, it is likely that Resh was  then pronounced as an alveolar trill, as it still is in Sephardi Hebrew.
There is still another pronunciation, affected by the addition of a dagesh in the Resh in certain words in the Bible, which indicates it was doubled [ʀː]: הַרְּאִיתֶם [haʀːĭʔiˈθɛm]. As can be seen, this pronunciation has to do with the progressive increase in length of this consonant (הָרְאִיתֶם). It was preserved only by the population of Ma'azya (מעזיה), which is in Tiberias.
A possible threefold pronunciation of Taw ת. There are three words in the Torah, Prophets, and Writings in which it is said that "the Taw is pronounced harder than usual". It is said that this pronunciation was halfway between the soft Taw ת /θ/ and the hard Taw תּ /t/: וַיְשִׂימֶהָ תֵּל [wajsiˈmɛhɔ‿θ‿tel]

Vowels
marginalThe vowel qualities /a e i ɔ o u/ have phonemic status: אָשָׁם הוּא אָשֹׁם אָשַׁם (Lev. 5:19) and אָשֵׁם 'guilty', אִם 'when' and אֵם 'mother'. /ɛ/ has phonemic value in final stressed position רְעֶה רְעִי רָעָה, מִקְנֶה מְקַנֵּה, קָנֶה קָנָה קָנֹה, but in other positions, it may reflect loss of the opposition /a/: /i/. By the Tiberian period, all short vowels in stressed syllables had lengthened, making vowel length allophonic. Vowels in open or stressed syllables had allophonic length (such as /a/ in יְרַחֵם, which was previously short).The Tiberian tradition possesses three reduced (ultrashort, hatuf) vowels /ă ɔ̆ ɛ̆/ of which /ɛ̆/ has questionable phonemicity. /ă/, under a non-guttural letter, was pronounced as an ultrashort copy of the following vowel before a guttural (וּבָקְעָה [uvɔqɔ̆ˈʕɔ]) and as [ĭ] preceding /j/, (תְדַמְּיוּנִי [θăðammĭˈjuni]). However, it was always pronounced as [ă] under gutturals: חֲיִי [ħăˈji].

Stress
Tiberian Hebrew has phonemic stress (בָּנוּ֫ /bɔˈnu/ 'they built' vs. בָּ֫נוּ /ˈbɔnu/ 'in us'). Stress is most commonly ultimate, less commonly penultimate, and rarely antepenultimate stress: הָאֹ֫הֱלָה /hɔˈʔohɛ̆lɔ/ 'into the tent'.

Phonotactics
As described above, vowel length is dependent on syllable structure. Open syllables must take long or ultrashort vowels; stressed closed syllables take long vowels; unstressed closed syllables take short vowels. Traditional Hebrew philology considers ultrashort vowels not to be syllable nuclei.

Orthography
The simple sheva sign changes its pronunciation depending on its position in the word (mobile/vocal or quiescent/zero) and its proximity to certain consonants.
In these examples, it has been preferred to show one in the Bible and represents each phenomenon in a graphic manner (a chateph vowel), but the rules still apply when there is only a simple sheva (depending on the manuscript or edition used).
When the simple sheva appears in any of the following positions, it is regarded as mobile (na):

At the beginning of a word, which includes the sheva (originally the first of the word) following the attached particles bi-,ki-,li- and u- and preceded by metheg (the vertical line placed to the left of the vowel sign, which stands for either secondary stress or its lengthening). Examples:  וּֽזֲהַב‎ /ˌʔuzăˈhav/ Genesis 2:12; בִּֽסֲבָךְ‎ /ˈbisăvɔx/ Psalms 74:5. But is not pronounced if there is no metheg; that is, they form a closed syllable.
The sheva following these three vowels /e/, /ɔ/, /o/, except for known types of closed syllables (and preceded or not, by metheg). Examples: נֵֽלֲכָה־נָּא‎ /ˌnelăxɔˈnːɔ/ Exodus 3:18; אֵ֣לֲכָה נָּא‎ /ˈʔelăxɔ ˈnːɔ/ Exodus 4:18.
The second of two adjacent shevas, when both appear under different consonants. Examples: אֶכְתֲּבֶנּוּ‎ /ʔɛxtăˈvɛnːu/ Jeremiah 31:33; וָאֶשְׁקֲלָה־לֹּו‎ /wɔʔɛʃqălɔˈlːo/ Jeremiah 32:9 (except for at the end of a word, אָמַרְתְּ‎ /ʔɔˈmart/).
The sheva under the first of two identical consonants, preceded by metheg. Examples: בְּחַֽצְצֹן‎ /băˌħasˤăˈsˤon/ Gen. 14:7; צָֽלֲלוּ‎ /sˤɔlăˈlu/ Exodus: 15:10.
The sheva under a consonant with dagesh forte or lene. Examples: סֻבֳּלוֹ‎ /subɔ̆ˈlo/ Isaiah 9:3; אֶשְׁתֳּלֶנּוּ‎ /ʔɛʃtăˈlɛnːu/ Ezekiel 17:23.
The sheva under a consonant that expects gemination but is not so marked, for example, the one found under ר‎. And sometimes even מ‎ when preceded by the article. Examples: מְבָרֲכֶיךָ‎ /măvɔʀăˈxɛxɔ/ Genesis 12:3; הַמֲדַבְּרִים‎ /hamăðabăˈʀim/ 2 Chronicles 33:18.
In case a quiescent sheva was followed either by a guttural or yodh, it would turn into mobile according to the rules given below, if preceded by a metheg. Ancient manuscripts support that view. Examples: נִֽבֳהָל‎ /nivɔ̆ˈhɔl/ Proverbs 28:22; שִֽׁבֲעַת‎ /ʃivăˈʕaθ/ Job 1:3.
Any sheva, if the sign metheg is attached to it, would change an ultrashort vowel to a short, or normal length vowel. For this, only ancient, reliable manuscripts can give us a clear picture, since, with time, later vocalizers added to the number of methegs found in the Bible.The gutturals (אהח"ע‎), and yodh (י‎), affect the pronunciation of the sheva preceding them. The allophones of the phoneme /ă/ follow these two rules:

It would change its sound to imitate that of the following guttural. וּקֳהָת‎ /ˌʔuqɔ̆ˈhɔθ/ Numbers 3:17; וְנִזְרֳעָה‎ /wănizrɔ̆ˈʕɔ/ Numbers 5:28.
It would be pronounced as ḥireq before consonantal yodh. Examples: יִרְמִיָהוּ‎ /jiʀmĭˈjɔhu/ Jeremiah 21:1; עִנִייָן‎ /ʕinĭˈjɔn/ in Maimonides' autograph in his commentary to the Mishnah.It must be said that even though there are no special signs apart /ɛ̆/, /ă/, /ɔ̆/ to denote the full range of furtive vowels, the remaining four (/u/, /i/, /e/, /o/) are represented by simple sheva (ḥaṭaf ḥiriq (אְִ‎) in the Aleppo Codex is a scribal oddity and certainly not regular in Hebrew manuscripts with Tiberian vocalization).
All other cases should be treated as zero vowel (quiescent, nah), including the double final sheva (double initial sheva does not exist in this Hebrew dialect), and the sheva in the words שְׁתַּיִם‎ /ˈʃtajim/ and שְׁנַיִם‎ /ˈʃnajim/, read by the Tiberian Masoretes as אֶשְׁתַּיִם‎ /ʔɛʃˈtajim/ and אֶשְׁנַיִם‎ /ʔɛʃˈnajim/ respectively. This last case has similarities with phenomena occurring in the Samaritan pronunciation and the Phoenician language.
Depending on the school of pronunciation (and relying on musical grounds, perhaps), the metheg sign served to change some closed syllables into open ones, and therefore, changing the vowel from short to long, and the quiescent sheva, into a mobile one.
That is referenced specifically by medieval grammarians:

If one argues that the dalet of 'Mordecai' (and other letters in other words) has hatef qames, tell him, 'but this sign is only a device used by some scribes to warn that the consonants should be pronounced fully, and not slurred over'.
The names of the vowel diacritics are iconic and show some variation:

The names of the vowels are mostly taken from the form and action of the mouth in producing the various sounds, as פַּתַ֫ח opening; צֵרֵ֫י a wide parting (of the mouth), (also שֶׁ֫בֶר) breaking, parting (cf. the Arab, kasr); חִ֫ירֶק (also חִרֶק) narrow opening; ח֫וֹלֶם closing, according to others fullness, i.e. of the mouth (also מְלֹא פּוּם fullness of the mouth). קָ֫מֶץ also denotes a slighter, as שׁוּרֶק and קִבּוּץ (also קבוץ פּוּם) a firmer, compression or contraction of the mouth. Segôl (סְגוֹל bunch of grapes) takes its name from its form. So שָׁלֹשׁ נְקֻדּוֹת (three points) is another name for Qibbúṣ. Moreover the names were mostly so formed (but only later), that the sound of each vowel is heard in the first syllable (קָמֶץ for קֹמֶץ, פַּתַח for פֶּתַח, צֵרִי for צְרִי); in order to carry this out consistently some even write Sägôl, Qomeṣ-ḥatûf, Qûbbûṣ.

Notes
References
Bibliography
Bar-Asher, M. (1998). Scripta Hierosolymitana Volume XXXVII Studies in Mishnaic Hebrew.
Blau, Joshua (2010). Phonology and Morphology of Biblical Hebrew. Winona Lake, Indiana: Eisenbrauns. ISBN 978-1-57506-129-0.
Dotan, A. (1967). The Diqduqe Hatte'amim of Aharon ben Moshe ben Asher.
Eldar, I. (1994). The Art of Correct Reading of the Bible.
Ginsburg, C.D. (1897). Introduction to the Massoretico-Critical Edition of the Hebrew Bible.
Golomb, D. M. (1987). Working with no Data: Semitic and Egyptian Studies presented to Thomas O. Lambdin.
Hayyim, Z. B. (1954). Studies in the Traditions of the Hebrew Language.
Janssens, Gerard (1982). Studies in Hebrew historical linguistics based on Origen's Secunda. Vol. 9. Uitgeverij Peeters. ISBN 978-2-8017-0189-8. {{cite book}}: |journal= ignored (help)
Malone, Joseph L. (1993). Tiberian Hebrew phonology. Winona Lake, IN: Eisenbrauns.
Sáenz-Badillos, Angel (1993). A History of the Hebrew Language. Cambridge University Press. ISBN 0-521-55634-1.
Steiner, Richard C. (1997), "Ancient Hebrew", in Hetzron, Robert (ed.), The Semitic Languages, Routledge, pp. 145–173, ISBN 0-415-05767-1
Yeivin, Israel (1980). Introduction to the Tiberian Masorah. Scholars Press. ISBN 0-89130-373-1.