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Comets - Comets are pieces of mostly ice, some rock, and organic compounds that can be up to several miles in diameter. They formed in the outermost parts of our solar system, beyond the orbits of our outermost planets - we get to see them because their orbits out there were disrupted. It is relatively common that a comet will come far enough into the center of the solar system that the ice gets vaporized by light from the sun and forms a kind of atmosphere called a coma, while the dust that was at the center, gets pulled out into a tail. A common misconception with comets is that their tail is opposite their direction of movement - like a tail of anything moving though our atmosphere would. However, the tail of a comet actually opposes the sun because it is the sunlight moving past he comet that pulls the tail out.
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Asteroids - An asteroid is an object that is usually composed of mostly rock - many have some amount of metal (mostly nickel and iron), but the amount varies. They are generally quite small but anything from a small bolder sized rock up to something hundreds of miles in diameter can be classified as an asteroid. Asteroids orbit the sun and really don't do much normally. They have very regular, predictable orbits and the vast majority stay within particular zones of orbit, called belts, that never intersect with any planets.
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Meteors - A meteor is what we commonly call a "shooting-star." A meteor is a very small meteorite that has entered Earth's atmosphere and relatively quickly burns up due to the speed it is moving at and the friction with the air with which it comes in contact.
Meteorites - A meteorite is a meteoroid that enters the atmosphere, but doesn't completely burn up as it enters. Any meteoroid that hits the ground is classified as a meteorite. These are generally the larger objects and are more often than not made out of more densely packed, often metallic material. Most meteorites that have impacted the surface of Earth have are believed to be fragments resulting from the collision of asteroids.