A comet is an icy lump of rock/ice that has its own orbit around the sun. They are kilometres in diameter, and may be regular (eg Halley's Comet every 76 years) or one offs, appearing from deep space, getting pulled towards the sun, and flung back off into oblivion.
A meteor is a small object (usually very small; a common shooting star is no bigger than a grain of sand) that enters the earth's atmosphere and burns up.
A meteorite is a meteor that hits the surface of the earth. Very few get this far.