An Asteroid, also called minor planet, is a member of a group of small, planet-like bodies that are part of our solar system. They are believed to be remnants of the nebula that were not incorporated into planets. One large group have orbits between the orbits of Mars and Jupiter, these couldn't form a planet due to the gravitational influence of Jupiter. Another important group is called Trojans; they are in the orbit of Jupiter, on two Lagrangian points. There is increasing interest in identifying asteroids whose orbit crosses the Earth's orbit, and that could, given enough time, collide with it. The two most important groups of near-Earth asteroids are the Amors, and the Atens.
The largest asteroid in the inner solar system is Ceres with a diameter of 1003 km. It also was the first to be discovered, by Giuseppe Piazzi on January 1, 1801. Nowadays, over 9000 asteroids are known, some less than 1 km across. Two other large asteroids are Pallas and Vesta. The first nearby pictures of an asteroid were taken by the Galileo spacecraft of Gaspra and Ida in 1991, while NEAR Shoemaker landed on Eros in 2001.
Another group of objects that didn't accrete to form planets (but are more icy, and so are not really asteroids) are the Kuiper Belt objects. They are the origin about half of the comets that come to the inner solar system. Some of these are not much smaller than Pluto and Charon - the largest found so far is 2001 KX76, thought to be around 1200 km around - and some astronomers expect that we shall one day find some Trans-Neptunian objects bigger than Pluto.
There are also a few objects that orbit the Sun between the oribts of the giant planets, called Centaurs. The first of these to be discovered was Chiron in 1977. These are generally supposed to be asteroids or comets that were ejected from their proper orbits.
When the orbit of an asteroid is confirmed, it is given a number, and later it may also be given a name (e.g. 1 Ceres). The first few are named after figures from Graeco-Roman mythology, but as such names started to run out, others were also used - famous people, the names of the discover's wives, even television characters. A few groups have names with a common theme - for instance Centaurs are all named after legendary Centaurs, and Trojans after heroes from the Trojan War. The Centaurs are of special interest; many of them are massive comets, such as Chiron.
The largest asteroids:
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Other noteworthy asteroids:
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Public domain picture from NASA