On a clear, very dark night, one may expect to see five to ten meteors per hour. These will come from all directions and most will be faint. If you aren't looking up at the instant one appears, you'll miss it. Better opportunities for observing lots of meteors present themselves during the strongest annual meteor showers. These showers occur when the Earth meets a dense stream of debris in orbit around the Sun. The debris originally came from a comet or asteroid. The Quadrantids, Eta Aquarids (best for observers near the equator and in the Southern Hemisphere), Perseids, and Geminids are usually the best annual meteor showers. They are named from the constellation or area in the sky where the meteors appear to originate. Meteors will be visible in any part the sky, but if you traced their paths backwards they will cross near a point called the radiant. Watching carefully during the peak of one of these showers, you may see a meteor every minute or so. More meteors will be seen if the Moon is not in the sky during the peak, and there are variations from year to year.
The Quadrantids originate from a radiant between the constellations Boötes and Draco. They are only at peak activity for a few hours, which can occur on January 3 or 4. If you look too early or too late, or if the maximum occurs during daylight hours, few meteors will be seen.
The Eta Aquarids peak on May 4. From northern temperate latitudes, morning twilight begins just as the radiant (in the constellation of Aquarius) gets high enough to start seeing a lot of meteors. In the Southern Hemisphere, twilight begins later in the morning, and the shower can be observed more easily.
The Perseids are a favorite shower, peaking in mid-August when the weather is warm for northerners. The rates are typically good on August 11 and 12, although variations in the time of maximum can lead to disappointment for those who are only impressed by lots and lots of meteors or live in locations where only the brightest ones are visible. In 1992 the Perseids' parent comet, Swift-Tuttle, returned to the inner solar system for the first time in over 130 years. There was speculation that a storm might occur in 1993 or 1994. The rates were high both years, with a peak of over 200 meteors per hour, but not nearly high enough to qualify as a real storm.
The Geminid shower peaks during the cold northern nights of December 13-14. If the moon is nearly new and the weather cooperates, you will usually see more meteors from this shower than from the others mentioned here.
There are other, less productive showers throughout the year. On occasion, a meteor storm will occur, usually associated with the close passage of one of the showers' parent comets. The Leonid shower usually produces about 15 meteors per hour on November 17-18, but in 1966 the rate was 150,000 for one hour! Comet Tempel-Tuttle, which supplies the Leonid dust, had just passed near the Earth and brought with it a denser portion of the meteoroid stream. Rates have increased each year since 1996, with spectacular fireballs reported each year and a peak rate of about 340 meteors per hour in 1998. In 1999, a brief storm was observed from Europe and Africa with rates of up to 70 per minute, but observers in other parts of the world saw comparatively few Leonids. Enhanced Leonid activity should be observed for several years to come, with some possibility of a meteor storm each year.
Meteor observing is simple: you need only your eyes, a bit of patience, and some warm clothes. Amateur astronomers can also make contributions to meteor science by carrying out formal watches. More information is available at these sites. A couple of reference guides for amateur meteor observers are the International Meteor Organization's Handbook for Visual Meteor Observers and Neil Bone's Meteors.