Sport climbing is the flavor of rock climbing in which the climber depends on fixed bolts for safety. These bolts are permanently drilled into the face of the rock, at intervals along a given climbing route. In general they are placed by the first ascensionist, and they are always left in place for the use of climbers who come later.
This is as opposed to "trad" or "traditional" climbing, in which each climber brings all his own protection, and takes it away again when he leaves. Sport and trad collectively are called lead climbing; the presence of bolts is what makes the difference between them.
Required gear for the sport climber includes a harness, a climbing rope, and a variable number of quickdraws. (A quickdraw or "draw" is a pair of carabiners connected by a flexible sling.) He also needs a partner, called a belayer, who is responsible for controlling the other end of the rope.
When preparing to ascend a sport route, the climber starts by attaching an end of the rope to his harness. He then climbs up several feet (perhaps 3 or 4 meters) until he can reach the first bolt. At that point he must stop, and "clip" the bolt: he takes out a quickdraw, attaches one end to the bolt, and the other end to the middle of the rope. Now the rope runs from his harness, up to the clipped bolt, and then down to the belayer.
Once clipped in to at least one bolt, the climber is protected. If he makes a mistake and falls off the rock, the rope will theoretically come taut in an inverted V between his harness, the bolt, and the belayer. This will halt his downward motion, preventing a painful impact with the ground.
As the climber passes additional bolts along the route, he clips into them as well, so that the rope runs down through a series of quickdraws before reaching the belayer. If the highest piece of protection fails to halt a fall for any reason, the next lower piece (again theoretically) remains connected and will be able to arrest his fall. His safety thus assured, the climber can fall as many times as he likes and still survive to reach the top of the route.
Note the "theoretically" qualifier up there. Unfortunately no system is perfect, and any mistakes here can lead to serious injury.
The most common mistakes in lead climbing involve bad clipping. For instance, a back clipped draw is one placed such that during a fall, the rope will wrap around it and pull the gate open. The open gate lets the rope pop out, meaning the climber is no longer attached to the bolt! Various other clipping errors can also allow a biner gate to open and the draw to become detached.
Another potential problem, specific to sport climbing, is in the bolts themselves. Since the climber does not place his own bolts, he is relying on the expertise of whoever did place them, and there's no guarantee that they can withstand any force. Note also that the bolts are permanent, and remain constantly exposed to the elements for years at a time. Even the best equipment will slowly rust and weaken, eventually to the point that it will break when subjected to a fall.
Of course, most bolts are bombproof and secure, and depending on them is as safe as lead climbing gets. But the climber is trusting each bolt with his life, and there is always the chance that a critical bolt will fail and drop him on his head. This is the tradeoff of climbing sport rather than trad. You gain the simplicity of not having to place gear yourself, at the expense of having to rely on what's been placed for you.