Asteroids are the rocky/dusty small bodies orbiting the sun. Meteorites are pieces on the ground left over after an asteroid breaks up in the atmosphere. Most asteroids are not strong, and when they disintegrate in the atmosphere they often produce a shower of meteorites on the ground.
Comets are also small bodies orbiting the sun, but they also contain ices that produce a gas and dust atmosphere and tail when they get near the sun and heat up.