Comets are usually made up of dust and ice. They grow "tails" when they come close to the sun. A nucleus is what comment have in common. These nucleus are usually ice or a rock.
Meteors are what we usually call shooting stars. They are these white and short trails that decorates the night sky during meteor showers. These trails are caused when they meteors burn up when they come close to the Earth's atmosphere at very fast velocities.