Q:

What is a Shivling?

As far as I know and understand, the Sanskrit word "lingam" लिन्गम्  means the genital organ of a human or an animal.
Is the Shivling the same thing? If this is so, what is the significance behind this portrayal and how did worshipping the Shivling become a common practice today?

A:

Etymology of Linga
Before entering into an explanation of Shiva Linga, I have to say a few words on the meaning of the word Linga. 
Linga is derived from the Sanskrit root Lika which means to sculpt or to paint, and Linga means one that sculptures or paints. God being the Sculptor of the Universe, He is known as Linga, and this word has become ultimately to mean any form or symbol that represents Him.
It has become in a later stage to mean any sign or symbol, in a general sense, and it is in this sense the word is used now. Refer to any dictionary, Sanskrit or Tamil and you will find the meaning of the word as a symbol or mark - a சின்னம், a குறி. The word is used in this sense by Lexicographers, Grammarians and Logicians; and it may even be found as a technical term used in this sense in Hindu Logic. Lingapattiam is the name of a commentary on the meanings of Sanskrit words, and one could clearly see in what sense the word is used as the title of that Book. But of course, in course of time, the word happened to convey other meanings as well and among them that of the generative organ, by the common law of degeneration of words which is not peculiar to Tamil or Sanskrit alone. Even then, this degenerated import of the word is not its chief meaning, but it is only a secondary one of very rare use. How the word happened to be used in this sense could itself be easily traced. There is an etymological rule in Tamil known as இடக்கரடக்கல் which is a form of decorum used in giving expression to objects which would not admit of open mention. The genital organ came to be referred according to this rule as இலிஙம் or குறி, and the use of these words in this sense has become a fashion in course of time. Not only the word Lingam but the word Kuri itself is used in this sense; but no one who has any idea of Tamil will contend that every Kuri is a genital organ. The word Kuri means punctuation, a brand mark, and I am sure that no punctuation will ever be said to represent a genital organ.
Another derivation, of the word Lingam is Ling, which means involution, and Gam, which means evolution. So, that Lingam is the principle of involution and evolution combined together, and such a combination can only be traced to God, the primordial cause of the whole Jagat.
It could thus be clearly seen that the radical meaning of the word Lingam does not in the least convey any sense applicable to the generative organ, but, on the contrary, the real meaning of the word may be found so sublime and so deep that it will immensely benefit one to scrutinize the word and learn its meaning analytically.
The Meaning of the Shiva Linga
So far for the word Lingam. I will now proceed to explain, as briefly as I can, the meaning of Shiva Lingam, but I must say at the outset that this object has a large stock of mysticism about it which can only be explained by an adept for whose qualifications I have the least pretence. Shiva Lingam is explained at great length by Shiva Agamas and several Puranas; and the Vedas themselves could be found to have their own explanation of Shiva Lingam.
It is the main principle of Hindu philosophy - and I am sure that principle is admitted by all religions -, that every particle of this great Jagat is moved by God. There cannot be any movement without God, and the existence of the universe would be altogether impossible without Divine energy. The principle of creation has been very minutely and systematically described in our sastras according to which Shiva Linga is the embodiment of the cosmic creation. There was the Nirguna Brahm; and there was the primordial cosmic element called Maya. What was the course taken by the Divine Energy in producing the cosmic world out of Maya? Maya is an extremely subtle matter without any form or shape, and it is of two kinds - Suddha Maya and Asuddha Maya - or the lower and upper Maya. This Maya is in the presence of Shivam or Nirguna Brahma and that of its Sakti or Divine Energy. This Sakti having energized Suddha Maya, the mundane egg of the universe was formed. This was Nadha or the principle of sound. This was what is known as Nama or name - the first expression of limitation. From this Nadha or Name came out Bhindu or Rupa i.e., the form - the second stage of limitation. This name and form - Nama and Rupa - is what is known as Omkara Pranava; and this is the seed and seat of all matter and force. The Nadha is represented by a line and the Bhindu by a disc. It is this Nadha or vibration that is known as Linga and Bhindu is what is known as its Pita. This Lingam with its Pitam or the principle of Name and Form is still beyond comprehension, and the form that could be comprehended a little better came out of the Bindhu above referred to in the order of evolution. This is what is known as Sadakkiam or SadaShivam. This is Rupa-Rupam, or with shape and without shape. From this SadaShivam came out Maheswara. With fully developed form, from him Rudra, in the region of Asuddha Maya, from him Vishnu, and from him, Brahma. These nine different phases or Navapitam are the different stages of evolution which the great God - or properly speaking - His Sakti - assumed in manifesting Itself to the souls - or in fact to excite their intelligence, and evolve this Jagat or universe out of Maya. The different actions in the region of Suddha Maya are performed by SadaShivam and Maheswara, while those in the lower Asuddha Maya by Rudra, Vishnu and Brahma - the Hindu triad. It could thus be seen what position the Nadha and Bhindhu hold in the order of cosmic evolution. These two principles as I have already said are known as Pranava - Nadha representing Nama, and Bhindhu representing Rupa - and it is this Pranava that is represented by Shiva Lingam. Nadham or the principle of egg - is represented by a line and Bhindhu, the next stage, by a disc. The line is the Linga and the disc is the Pita. We know that the principle of all writings in any language is embodied in this line and disc. Can we with any sense of correct knowledge call this Linga an emblem of generative organ? I am sorry that our critics are unable to form an idea of the creative principle except through the genital organ. You will see that in the order of evolution above out-lined, no fully developed form is manifested until the stage of Maheswara is reached.

Quoting S.Sabaratna Mudaliyar in Lecture on the Shiva Linga 

A:

This page explains,

Linga means "sign, mark, or symbol". It also refers to a digging
  implement or plough. Since the plough readies the earth for
  insemination, the word linga is also applied to the phallus,
  particularly as the phallic emblem of Shiva. Though Shiva is
  considered the destroyer (Trimurti) according to Hindu belief,
  creation (Srishti) follows destruction. The linga is therefore
  associated with the sexual union of Shiva and Parvati, which results
  in creation. It is a symbol of the procreative power of the universe.
  Linga is derived from the Sanskrit word lingam, which comes from li
  meaning, "to dissolve" and gam "to move on". This refers to the belief
  that one appears as a being in the world and then dissolves back into
  the universe.
A Shivalinga is phallus-shaped and fixed on a base, which is shaped
  like a yoni. The structure symbolises the supreme creative energy. It
  is usually made of stone but can also be of wood, metal, crystal, and
  soapstone.
There are various explanations as to why the linga represents Shiva.
  According to the Vamana Purana, after Sati's death, Shiva went to
  mourn in the forest. Seeing the virile god, the wives of the sages
  living in the forest were enamoured and followed Shiva around.
  Enraged, the sages cursed Shiva, causing his phallus to fall off. The
  moment it touched the ground, his phallus started growing. Soon the
  earth, unable to support its weight, began quivering. When Brahma and
  Vishnu realised why the earth was shaking, they requested Shiva to
  take back his phallus. Shiva consented to do so on the condition that
  the sages worshipped his phallus. He pronounced that nothing would be
  impossible for one who worshipped the Shivalinga.
The Linga Purana says that Brahma and Vishnu were once arguing over
  who was the supreme being. Suddenly, there appeared Sivalinga before
  them a huge column of fire. Both of them decided to find one end each.
  Whoever returned first would be acknowledged as supreme. Vishnu
  assumed the form of a boar and dug into the earth. Brahma, in the form
  of a swan, flew upwards. They searched for days but in vain. Then
  Shiva appeared in the fiery column. Brahma and Vishnu realised their
  mistake and acknowledged Shiva as the Supreme Being. The Shivalinga
  represents that column of fire Jyotirlinga
The phallic cult appears to have been prevalent in India since the
  Indus Valley Civilisation (c. 3000 BC - 700 BC). Emblems resembling
  the linga and yoni have been discovered in excavations of ancient
  cities. Although the Vedas are contemptuous of the linga because of
  its association with the phallus, later literature like the Puranas
  and tantric texts laud phallic worship. According to these texts,
  Brahma, Vishnu and other gods live at the site where Shiva exists in
  the form of a linga. The Vamana Purana says that during Chaturmasya,
  Shiva sleeps in all lingas. Therefore, praying at any linga during
  this time is considered especially auspicious. It is believed that the
  land within 100 cubits (about 160 ft) of a linga is very sacred and
  called Shivasthala.
There are approximately 30 million lingas in temples and shrines all
  over India. Most of them are named after the place where they are
  located or after the person who established them there. Lingas fall
  into three categories. The first are the Anadi or Svayambhu, or
  self-generated lingas. They are believed to be the most sacred and are
  said to have fallen from heaven. Next are the Sithavara, or fixed
  lingas. These are crafted and then, after an initiation ceremony done
  according to Puranic rites, installed at a sacred location. It is
  believed that once a linga is installed, it should never be moved as
  that would result in a calamity. The third category of lingas are the
  Jangam, or movable lingas. These are the miniature lingas, which are
  made of stone, clay, metal, or other material. They are worn as
  amulets and necklaces, or worshipped in the homes of Shiva's devotees.
The actual worship of the Shivalinga begins with bathing it, first
  with water mixed with earth, and then with water mixed with the
  Panchagavya. The linga is then smeared with sandalwood paste. Next,
  flowers and leaves are offered especially bel leaves. Next comes the
  arati, followed by the devotee's prayer. Shiva is believed to be very
  hot-tempered: bathing the linga and applying sandalwood paste are
  believed to soothe him.

