What causes a toilet to run constantly after flushing?

A constantly running toilet is almost always caused by a faulty flapper valve at the bottom of the tank. The flapper is a rubber seal that lifts when you flush and then drops back down to let the tank refill. If it's warped, worn, or has mineral buildup, it won't seal properly and water continuously leaks into the bowl, causing the fill valve to keep running. Other causes include a float set too high, which lets water rise above the overflow tube and drain constantly, or a worn fill valve that won't shut off. You can diagnose the flapper by putting food coloring in the tank; if color appears in the bowl without flushing, the flapper needs replacing.