Meteors (or shooting stars) are very different from comets, although the two can be related.
A Comet is a ball of ice and dirt, orbiting the Sun (usually millions of miles from Earth). As the ices in the nucleus are heated and vaporized by the Sun, gas escapes, taking dust particles along with it. The dust and gas are pushed away from the Sun by the solar wind and radiation pressure, producing the comet's tail (which always points away from the Sun, by the way). Large particles (the size of a grain of sand or larger) take a long time to be pushed around, so they remain in nearly the same orbit as the comet for months or even years, forming a large "cloud" of dust.
A Meteor on the other hand, is a grain of dust or rock (see where this is going) that burns up as it enters the Earth's atmosphere. The brief trail that is seen is the material that has burned off and is still glowing as it cools. Most meteors are only about the size of a grain of sand, and burn up completely in the atmosphere. Once in a while, however, a large meteor will survive and hit the ground (at which time it is called a meteorite).
A Meteor Shower takes place when the Earth crosses one of the "clouds" of dust that is in the comet's orbital plane. The duration and intensity of the shower is determined by the size and density of the cloud of dust. If the cloud is widely spread out, then the shower could be seen as a few meteors per hour for several days. On the other hand, if the cloud is very small and dense, then the shower will consist of thousands of meteors raining down in only a few minutes (which is classified as a Meteor Storm ).