Asteroids are large, rocky bodies from meters to kilometers in size often either 'stony' or 'iron-nickel in compositions with some rich in carbon compounds and classed as 'carbonaceous chondrites'. Most that strike Earth seem to be from the asteroid belt, with orbits perturbed by collisions among themselves, or distant orbit-changing encounters with Jupiter or Mars.
Comets are large kilometer-sized bodies rich in ices, and with it seems dust grains, gravel and small rocky bodies embedded in them. When the ices evaporate they grow tails and we see them as 'comets'. They leave behind in their orbits, gravel and rocky debris which the Earth may encounter from time to time as orbits cross, and we see these encounters as 'meteor showers'.
Meteorites are small rocky bodies, from micron-sized up to perhaps a meter across that impact the Earth after a brilliant atmospheric display. They can be fragments of material chipped off of asteroids that collided. They can be rocky cometary debris, or they can be just material in interplanetary space that over 4.5 billion years just never got incorporated into cometary bodies, planets or asteroids.