A shot of an Orionid meteor streaking across the sky this weekend. Source
Meteor showers are caused by the collision of the Earth with the field of debris left behind by a comet. As comets travel through their eccentric orbits they slowly break up, leaving behind trails of dust. Most of this dust is ice, but it is peppered with granules of more solid rock. When the Earth's orbit intersects these dust fields, the bits of rock smash into the atmosphere and burn brightly, creating the bright streaks that many call "shooting stars".
Comet Hyakutake shows off its tail. Source
Astonishingly, the bits of rock that create the brilliant light of a shooting star are usually only about the size of a grain of sand. These bits of dust are moving so fast in relation to the Earth that their impact with the upper atmosphere is explosive. Bigger meteoroids, say around the size of a small pebble, will make brighter streaks that may end in a bright flash. These are known as "fireballs".
Here a fireball is seen during an aurora over Norway. Source
Usually meteor showers are rather slow, with something like 20 to 30 meteors an hour, but sometimes they can produce more than 1,000 meteors an hour. When this happens, they are no longer referred to as "meteor showers", instead their called "meteor outbursts" or "meteor storms". These spectacular events don't happen very often, but we may see one again in 2014.
Leonid meteor storm. Source
One final interesting fact about meteor showers: they can appear anywhere there is a mostly transparent atmosphere. This means that planets like Mars also have meteor showers. Mars' thin atmosphere doesn't stop it from creating shooting stars when it runs into a comet's tail. The reason for this is fairly simple. On Earth, meteoroids burn up in the very upper atmosphere, where it's still thin, this means that Mars' atmosphere is thick enough to burn up these small particles.
Meteor over Mars. Source