The first thing you need to know is, it's awesome. I tried it while in New Zealand last month, and speaking as someone who's got issues with heights, I can assure you that once you've hurled yourself off the cliff/mountain/ledge and are airborne, it's just about the most fun you're likely to have. (As proof, check out this video of my flight and my wife's landing; we're practically giddy.)
1. It's not to be confused with hang-gliding
I found this out literally as I was being driven up a hair-raising mountain switchback to the launch point. (I recommend finding out a little earlier.) The major difference lies in the wing shape and design. Hang gliders are solid wing structures, utilizing an aluminum frame to create a V-shaped wing that resembles the stealth bomber. Paragliders are soft wing structures, with no internal frame, which once inflated have an elliptical shape. Because they have a slower flying speed, they're much more forgiving than hang gliders, and as a result the learning curve is usually less steep for paragliding. Also, paragliders fold conveniently into a small bag, so you can take them mountain trekking then paraglide down when you're tired of being on a mountain. (Link.)
2. It's also not to be confused with parasailing
Parasails are essentially just parachutes, and are used by people being towed by boats at the beach or on lakes. They never get more than a few hundred feet high at most. The design of a paraglider wing is more like that of a 747 than a parachute -- it's designed to catch thermal updrafts and rise through the air, not just fall slowly to the ground. It's a much more dynamic experience, which is why glider pilots are called pilots, and people who use parachutes aren't really called anything.
3. It's not as dangerous as it sounds
Before I tried it, I thought paragliding was one of those things that only suicidal maniacs and adrenaline junkies attempted. I am neither. It's actually one of the safest airborne sports. Firstly, you're connected to the wing by at least 30 lines, any one of which is strong enough to support your weight. There is a risk of the wing deforming and/or collapsing while in flight, but this is rare and usually due to pilots unwisely deciding to fly in bad weather. If you're at a sufficient altitude (more than 700 feet or so), the reserve parachute most glider pilots wear will guide them safely to the ground; the danger is to be too close to the ground when performing some dangerous maneuver, in which case your parachute won't have time to inflate properly before you splat back to earth.
4. It was named by NASA
Leonardo Da Vinci may have designed the first parachute, but NASA helped design, and name, the paraglider. In 1961, a French engineer named Pierre Lemoigne took the first steps by cutting strategically placed vents in a parachute which allowed it to ascend into the air and be steered, but it was NASA who developed what was known as a "sail wing," for use in recovery of lunar capsules, into the paraglider.
5. It's more comfy than the chair you're sitting in right now
Well, probably. Unlike most harness-based activities, like rappelling and parachuting, the focus isn't on a series of pinchy straps and clips around your legs and midsection. Modern paragliding harnesses connect you to something akin to a lounge chair, in which your reserve parachute and other goodies are stored, and some of them even feature lumbar support. After you leap from the precipice of choice and the glider inflates, you simply slide the "chair" under your bum, and ride your flying barcalounger wherever the wind takes you.
6. You've got brakes and gas
Well, sort of -- there's certainly no combustion engine on board, but pilots do have a great deal of control over their gliders. Controls held in each of the pilot's hands connect to the trailing edge of the left and right sides of the wing, and these can be used to steer and to adjust speed. The only thing directly under their control is ascent -- that depends on their skill (and luck) at finding rising columns of thermal air, which can loft the glider great distances.
7. You can travel across the country this way
If you're really, really good, that is. Most paraglider flights last between 15-25 minutes, depending on weather conditions. But pilots who are especially skilled at finding and exploiting thermal columns of rising air can use them to hop and skip their way across long distances, like paragliding champ Will Gadd, who holds the world record for longest paraglider flight, at 263 miles.