A meteor, or its common name shooting star, is when a piece of space debris enters the Earth's atmosphere. The streak of light we see on earth is the material that is being heated in the upper atmosphere. While some call this force creating the heat friction, it actually comes from ram pressure, the pressure on an object moving through a flowing medium. The streak of light is in the opposite direction in which the meteor is traveling.
A simple experiment of this is you holding your hand under a running faucet. The force on your hand by the water is the ram pressure. It's similar when you're wading across a stream or dip your hand into a river. The pressure is directed with the flow of the fluid.
Meteor showers are comprised of objects that burn up in the atmosphere, and an aerial explosion like what happened in the sky over Chelyabinsk was from a meteor.
While out in space, the proper term is "meteoroid". A good trick to remember is that it is a meteor in the void of space: meteor + void => meteoroid. These are objects that are significantly smaller than asteroids.
Now if a meteor survives its trip through the atmosphere and impacts the earth's surface, it is called a meteorite. A trick to remember, but not perfect: meteor + strike => meteorite. There is no such term as an "asterite", but asteroids that impact the surface are also termed "meteorites".
Comets are a little different due to their composition, but there will be two that will be visible from earth this year. Those will be explored in an article published later this week.