Unlike the other small bodies in the solar system, comets have been known about for thousands of years. There are Chinese records of Comet Halley going back to at least 240 BC. The famous Bayeux Tapestry, which shows the Norman Conquest of England in 1066, has a picture of Comet Halley.
As of 1995, 878 comets have been cataloged and their orbits at least roughly calculated. Of these 184 are periodic comets (they have an orbit of the sun of less than 200 years).
Comets are sometimes called dirty snowballs or "icy mudballs". They are a mixture of ices (both water and frozen gases) and dust that for some reason didn't get made into planets when the solar system was formed. This makes them very interesting as samples of the early history of the solar system.
When they are near the Sun and active, comets have several distinct parts:
Comets are invisible except when they are near the Sun. Most comets have highly eccentric orbits which take them far beyond the orbit of Pluto; these are seen once and then disappear for thousands of years. Only the short- and intermediate-period comets (like Comet Halley), stay within the orbit of Pluto for a significant fraction of their orbits.
After 500 or so passes near the Sun off most of a comet's ice and gas is lost leaving a rocky object very much like an asteroid in appearance. (Perhaps half of the near-Earth asteroids may be "dead" comets.) A comet whose orbit takes it near the Sun is also likely to either hit one of the planets or the Sun or to be ejected out of the solar system by a close encounter (esp. with Jupiter).
By far the most famous comet is Comet Halley but SL 9 was a "big hit" for a week in the summer of 1994.
Meteor showers sometimes occur when the Earth passes through the orbit of a comet.
Many comets are first discovered by amateur astronomers. Since comets are brightest when near the Sun, they are usually visible only at sunrise or sunset.