In the field of astronomy a centaur is a small body that orbits the Sun with an orbit extending past Jupiter, but not so far out as Neptune, and which has an orbit that crosses (or has crossed) the orbit of one or more of the giant planets. Or, to put it a bit more plainly, centaurs are a group of asteroids and comets that have unstable orbits due to passing too close to a large gas giant.
Some of the centaurs are icy enough that they exhibit a cometary tail when they are near to the sun, and thus are classified as comets. Currently the division between comets and asteroids is somewhat blurred, as historically the presence of a tail was the primary criteria; this classification has proven insufficient, as the appearance of tails is dependent on proximity to the sun, and thus not relevant to objects that do not approach the sun closely. It is expected that most, if not all, centaurs would exhibit a tail if they passed close enough to the sun.
Centaurs can be quite large, and it is interesting to note that Chariklo and Chiron (232 km and 206 km in diameter, respectively), are the only known bodies in the Solar System to have rings, other than the four gas giants. While the debate over what exactly constitutes a dwarf planet continues, by some measures the centaurs Chariklo, Chiron, and Bienor would be included.
Because the centaurs have unstable orbits, it is expected that over the course of tens of millions of years many centaurs will eventually stabilize into more regular orbits, resolving into either Kuiper belt objects, periodic comet, or Jupiter family asteroids. Likewise, many centaurs will collide with the sun or a gas giant, or simply be ejected into interstellar space. Not all centaurs need be so unfortunate; it is suspected that Saturn's moon Phoebe is a captured centaur.
As you might expect, the astrological centaurs are usually named after mythological centaurs. Additionally, the International Astronomical Union has decided that small bodies that pass beyond the orbit of Neptune (Trans-Neptunian object, or TNOs), while not qualifying as centaurs, should also be named after mythical centaurs if they have unstable orbits like the centaurs. This has resulted in these objects sometimes being included as centaurs as a class. Most suggested formal terminology does not technically classify these as centaurs despite the naming convention.