The ferryman of the dead, and a minor deity, Charon took properly buried souls1 - for the small price of an obol - from the shore of Cocytus over the river Styx to the rest of Hades, on which side they would be placed in their relevant areas: Tartarus for the wicked; Elysium for the good; and various other areas depending on whose vision of the Underworld you subscribe to.
Heroic Incursions
Apart from the drudgery of shifting souls around, Charon also had a duty to make sure no mortal ever crossed the Styx. Unfortunately, he was notoriously unreliable in this respect, causing a number of Greco-Roman heroes to take advantage and go on a tourist excursion.
Orpheus charmed the grim boatman with his lyre and managed to enter and resurrect his wife Eurydice. Hercules was charged to enter Hades as his last labour, and intimidated the senile sailor into letting him cross. The brute. Aeneas visited Hades for similar reasons, and was able to gain passage by the display of divine will which was his ability to obtain the Golden Bough.2 For his leniency toward Hercules, Charon was imprisoned by Hades (Pluto in Roman mythology) for an entire year by being chained to his oar. Which makes one wonder what the souls did while they waited.
The spectre himself
Charon's name almost certainly draws from Acheron, one of Hades' five rivers. He was the son of Erebus and Styx.3 He is universally described as wearing grimy clothes, having fiery eyes and being generally typical of the Underworld.
Probably the finest description, however, is in the Aeneid, in which he is given very human characteristics, perhaps constituting an excuse for eventual leniency toward Aeneas. He is said to be old, but strong, with a grey beard on his chin. His speech is nostalgic, as befits an old man. He is easily swayed by the stronger character of the Sibyl when he puts up an initial resistence to the thought of carrying Aeneas.
Charon's influence
The notion of a grim ferryman, named or not, is very widespread, with the most enduring image in the modern psyche being the skeletal figure in the film Clash of the Titans, conveying the souls across the Styx mechanically and haunting with his flaming eyes.
In contemporary Greek culture, the spirit of Charon lives on as Charontas, the Angel of Death. He is also to be found in Dante's Inferno, a work inspired by Greco-Roman mythology whose influences in modern literature are myriad.
1 Those who were improperly buried had to wait over a hundred years, which probably wasn't so bad for those destined to go to Tartarus after this time.
2 Odysseus also visited the Underworld, but no mention is made in Homer of Charon. It is in Vergil's inclusion of the ferryman that we see not only his debts to other writers and myths, but also his improvement on Homer's Odyssey. Although much of the Aeneid's sixth book draws from the Odyssey, much is added from other authors to give a much more vivid and moving account of Aeneas' visit as a worthy centrepiece to the whole work.
Theseus also visited the Underworld with Pirithous to save Persephone. No mention is made, however, of Charon in the original story.
3 Don't ask me how a river and the personification of darkness can have a child. They just could, OK?
Sources:
The Aeneid
http://www.dl.ket.org/latin1/mythology/1deities/underworld/charon.htm
Various nodes on E2