The flapper is the most common, and easily remedied, source of toilet trouble. It's a bulb, typically made of rubber, which hooks onto the fill valve in the toilet's tank and covers the hole leading from the tank to the bowl.
The top of the bulb has a chain which attaches to the flush lever, so that flushing the toilet pulls the flapper upward and lets the water in the tank enter the bowl, where it does what it's supposed to do. (unless the toilet is clogged, which is another problem entirely...)
Sometimes the flapper will come loose, in which case water will continue to leak into the bowl while the fixture is not in use. Returning it to its proper position is simple; you may need to remove a link or two from the chain to keep it more tightly secured. If the flapper appears to be in place and it's still letting water through, there is either a problem with the valve itself, or with the seal between flapper and gasket. Since the flapper is made of rubber and decays far more quickly than just about any toilet component, odds are good it's what needs to be replaced. A new one will cost you 2 or 3 US dollars.