A shooting star or meteor is a piece of natural debris - a chunk of rock and/or ice - that has been floating around in space for hundreds of millions of years, more or less. When one of these strikes the atmosphere of earth, it burns up, and we see it's light as a shooting star. Most burn up in the atmosphere. If a piece of one makes it to the earth, we call it a meteorite. Meteorites usually look like small, odd rocks.
Most meteors come from one of three places. The asteroid belt, between the orbits of Mars and Jupiter, is full of rocks. When two collide, one may be pushed our way. If it hits the atmosphere, it becomes a meteor. Other rocks and ice can come from the Kuiper belt, which is way out beyond Pluto. And a few come from the Oort Cloud, which is even farther out than that.
Meteors usually shine for just a few seconds before they burn up.
Comets are very different. They come from the Kuiper Belt or the Oort cloud, but are much larger, and do not strike the Earth. They shine as they are warmed by the sun and leave a trail of sparkling water across the heavens, and are visible for many nights as they pass the Earth, swing around the sun, and head back out into the far reaches of the solar system.