Description
Descriptions of the unicorn vary. The physician Cteisias of Cnidus, in his Indica, a history of Persia, asserts:
"There are in India certain wild asses which are as large as horses, and larger. Their bodies are white, their heads dark red, and their eyes dark blue. They have a horn on the forehead which is about a foot and a half in length. The dust filed from this horn is administered in a potion as a protection against deadly drugs. The base of this horn, for some two hands' breath above the brow, is pure white; the upper part is sharp and of a vivid crimson; and the remainder, or middle portion, is black. Those who drink out of these horns.... indeed, they are immune even to poisons..."
Later writers expand upon Cteisias' depiction, some asserting that the horn was a cubit and a half in length (Aelian). In the King James version of the Bible, several references are made to the "unicorn", though this is heavily dependent on the translation. (I must add, however, the NIV version of the Bible replaces "unicorn" with "wild ox", which, I suppose, lends a bit more credulity to the story as well as make it easier to translate such a phrase internationally.)
Over time, its appearance has changed drastically from this original probably due to Christian influences, Renaissance liberties, and the Millefleur tapestries (also more commonly known as the Unicorn Tapestries), though where the Millefleur tapestries owe their inspiration from, no one knows. Current day interpretations of the unicorn make it out to be a dainty, pure white horse with a white mane (albeit smaller than a horse), a spiraled slender horn reminiscent of the narwhal's horn (its very spirals are suggested to come directly from the narwhal), cloven hooves, and a beard.
Symbolism
Most people are familiar with the procedure of capturing a unicorn: that is, use a virgin girl and the unicorn will come and lay its head in her lap (a picture depicted very prettily in one of the Millefleur Tapestries, for example).
Medieval sensibilities were in conflict over this, interpreting the symbolism in two vastly different ways. The interpretation that has survived to this modern age, however, have been prone to allegorize the unicorn as some sort of symbolic interpretation of Christ, with the virgin as the Virgin Mary. The unicorn's characteristics make it especially well-suited for this allegory: the supposed chastity of the unicorn, its uniqueness, its self-sacrifice (in a sense), the properties of its horn, and most especially, its color - white being associated with purity. (How such a color change came about, considering Cteisias' initial interpretation, is clearly a chicken and egg deal; did the color change first to associate readily with interpretations of Christ, or did its connections to Christ cause the color change? Even Shepard passes by this change of appearance with very little examination or conclusion.) Either way, it is a curious interpretation, especially when you consider that the virgin girl who ensnares the unicorn is there explicitly to keep the unicorn unaware of the hunters, who are (presumably) hiding in ambush to slaughter the unicorn. And no matter how you look at it, it's rather hard to put mind to the idea that the Virgin Mary definitively lured Jesus to the slaughter, a Judas goat in her innocence.
True to medieval sensibilities, yet again, the Christians solved this problem by piling even more symbolism on top of the mess, employing a tortured logic that defies, well, logic. An old German story that tries to solve this knotty problem presents it thusly: A king had two sons, one of whom committed suicide, and the other of whom fell ill. As the king was anxious to cure his only remaining son, he found, through means of inquiry, that only the unicorn's blood may save him. And so he sets out to hunt the unicorn, procuring the most beautiful maiden in the kingdom (as well as six lovely companions) and setting his four dogs and hunters to this task. They succeed in their task, and the son is saved. Now, the writer of this story asserts, if you look at it symbolically, even spiritually, the representations are as follows: the king is God; his dead son is Lucifer; his dying son is Adam; the virgin is the Mary, her companions are personifications of virtues, the hunter is the Holy Ghost (sometimes in the form of the angel Gabriel), and the four swift dogs are, respectively Veritas, Justitia, Pax and Misericordia (respectively, Truth, Justice, Peace, and Mercy).
Whether or not this solved the problem is another matter entirely, but is really very inconsequential as the medieval ages as a whole can be considered an age of inconsistency, anyhow.
The other medieval interpretation, however, goes the opposite direction: the unicorn, indeed, is some sort of representation of the Devil, an evil that can only be overcome by good. This interpretation fits especially well when one reads the text of the Physiologus, a bestiary:
"There is an animal, called dajja, extremely gentle, which the hunters are unable to capture because of its great strength. It has in the middle of its brow a single horn. But observe the ruse by which the huntsman take it. They lead forth a young virgin, pure and chaste, to whom, when the animal sees her, he approaches throwing himself upon her. Then the girl offers him her breasts, and the animal begins to suck the breasts of the maiden and to conduct himself familiarly with her. Then the girl, while sitting quietly, reaches forth her hand and grasps the horn on the animal's brow, and at this point the huntsmen come up and take the beast and go away with him to the king."
Though the bestiary asserts a need for a virgin, later stories mention that if no virgin was available, any woman would do; indeed, if that was not even possible, Johannes Tzetzes (a Greek grammarian) remarks that a boy would do, as long as he was dressed as a girl and heavily perfumed (which must be some form of bastardization from the original, which asserted that the girl must be naked, and, on top of that, tied to a tree).
The fact that the unicorn did not seem too picky about the whole girl/boy deal led some to propose that the unicorn was drawn to the virgin because of the smell of innocence (or some semblance of it, bottled up in perfume), or through some sort of "humours" that the unicorn lacked but desired greatly. Later interpretations explained that the unicorn was drawn instead to the sight of such beautiful ladies, beardless and gentle, and would indeed be drawn out more successfully if several naked ladies were cavorting the forest, an explanation especially favoured by the Abbess Hildegarde.
Either way, the sexual connotations of this Syrian bestiary interpretation were obviously too much for the Christian sensibility, and were promptly buried, much like Perrault's original fairy tales.
Function
Aside from being an allegory of Jesus or the complete opposite, the capture of the unicorn was important particularly because of its horn, which was said to detect all poisons, and indeed, even purify if need be (the stories vary as wildly as the descriptions). The horn is also noted for destroying particularly dangerous creatures of poison, such as scorpions. The tests involved in determining a true unicorn horn are numerous, and quite bizarre, as the case of David de Pomis' belief:
"The test is this: place the horn in a vessel of any sort of material you like, and with it three or four live and large scorpions, keeping the vessel covered. If you find four hours later than the scorpions are dead, or almost lifeless, the alicorn is a good one, and there is not money enough in the world to pay for it. Otherwise, it is false."
Poison was the greatest fear of a politician, as news of the Borgias and the very art itself had exploded into a mythological frenzy, where a skilled poisoner could bypass a taste tester completely and kill through a poison that was diluted sufficiently enough to time death to hours, perhaps even days later. Naturally, enormous sums were paid for the powder of a horn.
Origin and Confusion
A great many unicorns seem to have come from the court of Prester John (sometimes referenced as Prestor John), a mythological priest and king who apparently held reign over a Christian kingdom (early on, in India; in later times, apparently Ethiopia) in a region full of heathens, specifically Muslim (a myth that I'm sure took root from the Crusades). He was a much remarked phantom, partly because of his apparent connections to the Pope, and partly because his kingdom seemed rather like the ultimate utopia, as he "holds full great land, and hath many full noble cities and good towns in his realm and many great diverse isles and large... (he) hath under him many kings and many isles and many diverse folk... and this land is good and rich... The Emperor Prester John is Christian, and a great part of his country is also." (I must note that Umberto Eco wrote a marvelous story about Prester John in his recent book Baudolino, if anyone wishes to read it.)
Aside from Prester John's magical horde of unicorns, there were numerous false "sightings", though in many cases, were in fact simply sightings of rhinoceros, which also had a similar "capture" legend affiliated with it (starting with "take two virgins in the wild and captivate the rhino..."; you get the rest). Marco Polo, recently returned from India, was clearly disappointed at the unicorn he saw, writing that it "is not in the least like that which our stories tell of as being caught in the lap of a virgin; in fact, 'tis altogether different from what we fancied."