A meteor shower is when the Earth passes through a field of debris, which burns up in the atmosphere, creating the familiar "shooting stars". The biggest difference, though, between this and normal meteor falls, is that during a shower there can be from 15 to several hundred meteors that can be observed each hour, whereas normally meteors are visible at a rate of two or three per hour.
Most of these meteors are caused by particles not much larger than a grain of sand. As these particles make contact with the Earth's atmosphere at several thousands of miles an hour, they heat up by friction, and briefly flare up, to become visible to those on the ground below.
Earth normally encounters thousands of tiny particles daily as it moves through it's orbit around the sun. Most of these are even smaller than in the case of a meteor shower; perhaps dust sized. Only those the size of a grain of sand up to that of a fist sized rock, produce the meteor shower we observe in the night sky. Note that those particles that are larger than a fist have the potential for making land fall. When this happens, they are no longer called meteors, they are called meteorites.
There are other objects, though, that also move around the sun, but in paths that intersect Earth's orbit, such as comets. Comets are composed of primarily dust, rock, and ice, and as they orbit the sun and are heated by the solar wind, bits, flakes, or particles break off of them. When the Earth then encounters these particles, there is a meteor shower, simply because of the sheer number of rocky bits. Since comets generally follow the same path each time around the sun, very often those that have been recorded before leave debris along the same trail, so astronomers can predict when the next occurrence of that particular meteor shower will be. And if the comet has been around the sun recently, since there will be more particles, there will usually be more meteors per hour, creating a much more spectacular display.
Usually, because of the position in space relative to the Earth as our planet plows through a debris trail, the meteorites often appear to come from a single quadrant or area of the sky. This makes it not only easy for the observer, who can then simply concentrate on looking at that one area, it also makes it handy for scientists charged with naming particular meteor showers, since they can simply reference the constellation it appears to come from. So for instance, the Leonids will appear to originate in the constellation of Leo, while the Aquarids seem to come from the constellation of Aquarius. Naturally, it is helpful for the observer to be able to identify the constellations, however there are quite a number of star charts that have this information.
For the remainder of this year, the major showers are the Aquarids (July 27, best viewed about an hour before dawn), the Perseids, seeming to come from the constellation of Persius (August 12 & 13), the Orionids, seeming to come from Orion (October 21...this is usually one of the best showers), the Leonids (November 18), and the Geminids, seeming to come from Gemini (December 14). Note that the dates given are for the peaks. People can usually see the storm a day or two in advance to a day or two after these dates. Also note that many of these showers are not visible from the Southern Hemisphere, which has it's own showers.
Meteor showers are natures own fireworks display. I hope that everyone will try to take the time to enjoy them. So pop up some popcorn, get out the lawn chair, relax, and enjoy the display that nature has already set up to give you!