An asteroid (minor planet or planetoid) is a relatively small "planet-like" object which orbits within a solar system; generally less than 1km in diameter. They are believed to be remnants of the protoplanetary discs; which were not incorporated into planets during the system's formation. Most asteroids will align themselves within an asteroid belt and some have moons; asteroids (like most celestial objects) tend to have elliptical orbits. "Small" asteroids (generally fragments from collisions) which are approximately boulder-sized or less, are known as "meteoroids". Asteroids composed largely of ice are known as comets. Some asteroids are the remains of old comets that have had most of their ice driven away by close proximitiy to the Sun, leaving behind only the heavier mineral components.
More than 9000 asteroids have been discovered within the Earth system. The largest asteroid in Earth's inner solar system is Ceres, with a diameter of 900-1000km. Two other large asteroids are Pallas and Vesta; both have diameters of ~500km.
A large group of asteroids have orbits between the orbits of Mars and Jupiter, roughly 2 to 4 AU, in a region known as the Main belt. These couldn't form a planet due to the gravitational influence of Jupiter. Jupiter's gravitational influence also results in Kirkwood gaps in the asteroid belt, orbits cleared by orbital resonance.
Another group exists within the Kuiper belt. The Kuiper belt is the source of 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 Quaoar, thought to be around 1250 km in diameter, the same size as (or even slightly larger than) Charon - and some astronomers expect that we shall one day find some Trans-Neptunian objects bigger than Pluto.
The Centaurs orbit the Sun, between the orbits of the giant planets. The first of these to be discovered was 2060 Chiron in 1977. These are generally supposed to be asteroids or comets that were ejected from their proper orbits.
Up until 1998, asteroids were discovered by a four-step process. First, a region of the sky was photographed by a wide-field telescope. Pairs of photographs were taken, typically one hour apart. Multiple pairs may be taken over a series of days. Second, the two films of the same region were viewed under a stereoscope[?]. Any body in orbit around the sun would move slightly between the pair of films. Under the stereoscope, the image of the body would appear to float slightly above the background of stars. Third, once a moving body was identified, its location would be measured precisely using a digitizing microscope. The location would be measured relative to known star locations.
These first three steps do not constitute asteroid discovery: the observer has only found an apparition[?]. The final step of discovery was to send the locations and time of observations to Brian Marsden[?] of the Minor Planets Center[?]. Dr. Marsden has computer programs that compute whether an apparition tied together previous apparitions into a single orbit. If so, then the observer of the final apparition is declared a discoverer, and the discoverer got the honor of naming the asteroid (subject to the approval of the International Astronomical Union).
Since 1998, a large majority of the asteroids have been discovered with automated systems that comprise CCD cameras and computers directly connected to telescopes. A list of teams using such automated systems include (http://neo.jpl.nasa.gov/programs):
The LINEAR system alone has discovered 157,920 asteroids as of April, 2002  (http://www.ll.mit.edu/LINEAR). Between all of the automated systems, 2298 near-Earth asteroids have been discovered  (http://neo.jpl.nasa.gov/missions/stats).
There is increasing interest in identifying asteroids whose orbit crosses the Earth's, and that could, given enough time, collide with the earth. The two most important groups of near-Earth asteroids are the Amors, and the Atens. Various asteroid deflection strategies have been proposed.
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.
Asteroids are classified into spectral types by their optical spectrum, which corresponds to the composition of the asteroid's surface material. Note that the proportion of known asteroids falling into the various spectral types does not necessarily reflect the proportion of all asteroids that are of that type; some types are easier to detect than others, biasing the totals.
There are also a number of rarer asteroid types, the number of types continuing to grow as more asteroids are studied.
The largest asteroids:
|Number||Name||Diameter (km)||Date Discovered||Discoverer|
|1||Ceres||1003||January 1, 1801||Piazzi, G.|
|2||Pallas||608||March 28, 1802||Olbers, H. W.|
|4||Vesta||538||March 29, 1807||Olbers, H. W.|
|10||Hygeia[?]||450||April 12, 1849||de Gasparis, A.|
|31||Euphrosyne[?]||370||September 1, 1854||Ferguson, J.|
|704||Interamnia[?]||350||October 2, 1910||Cerulli, V.|
|511||Davida[?]||323||May 30, 1903||Dugan, R. S.|
|65||Cybele||309||March 8, 1861||Tempel, E. W.|
|52||Europa||289||February 4, 1858||Goldschmidt, H.|
|451||Patienta[?]||276||December 4, 1899||Charlois, A.|
|15||Eunomia[?]||272||July 29, 1851||de Gasparis, A.|
|16||Psyche||250||March 17, 1851||de Gasparis, A.|
|48||Doris[?]||250||September 19, 1857||Goldschmidt, H.|
|92||Undina[?]||250||July 7, 1867||Peters, C. H. F.|
|324||Bamberga[?]||246||February 25, 1892||Palisa, J.|
|24||Themis||234||April 5, 1853||de Gasparis, A.|
|95||Arethusa[?]||230||November 23, 1867||Luther, R.|
Other noteworthy asteroids:
|Number||Name||Diameter (km)||Year Discovered||Comment|
|243||Ida||56 x 24 x 21||September 29, 1884||Visited by Galileo probe|
|Dactyl||1.4||1991||Moon of 243 Ida|
|253||Mathilde||66 x 48 x 46||November 12, 1885||Visited by NEAR Shoemaker|
|433||Eros||13 x 13 x 33||August 13, 1898||Visited by NEAR Shoemaker|
|624||Hektor[?]||February 10, 1907||Largest Jovian Trojan asteroid discovered|
|951||Gaspra||19 x 12 x 11||July 30, 1916||Visited by Galileo probe|
|2060||Chiron||170||1977||First Centaur to be discovered|
|4593||Reipurth||March 16, 1980||discovered by Claes-Ingvar Lagerkvist[?]|
|3753||Cruithne||5||October 10, 1986||Unusual Earth-associated orbit|
|4179||Toutatis||4.5 x 2.4 x 1.9||January 4, 1989||Will approach Earth closely in 2004|
|4769||Castalia||1.8 x 0.8||August 9, 1989||First asteroid to be imaged|
|5261||Eureka[?]||June 20, 1990||First Martian Trojan asteroid (L5 point) discovered|
|2002 AA29[?]||0.1||January 9, 2002||Unusual Earth-associated orbit|
Public domain picture from NASA