Have you ever flushed the toilet, only to go back to the bathroom an hour later and find it still running?  Do you live alone, but wake to the sound of water running in the middle of the night?  It's not ghosts, it's your toilet's flapper valve.

Toilets are fairly simple.  They are far less complicated than your TV, for instance.  Still they are more complicated than the kitchen sink.  When it works normally, the flush handle opens the flapper flooding the bowl and sending your deceased goldfish down the drain.  How Stuff Works has a great cutaway diagram of a toilet, with flush-able animation.
Most times when a toilet just keeps on running, it is a simple matter of something stuck under the flapper keeping it from closing.  Usually that something is the pull chain from the flush handle, and jiggling the handle shakes it loose.  The long term solution is to shorten the chain until it has as little slack as possible.  Sometimes the flapper will get old and harden so it will not seal very well, but they are easy to replace, cheap and nearly a universal fit.

The toilet that runs at night at random intervals has a different cause, and is a little harder to fix.  Typically when the tank fills to the pre-set level the float closes the valve shutting off the water supply.  When you have a float that doesn't float, a shut-off valve that doesn't shut off, or a flapper that has a slow leak, the water will run periodically for a few seconds then shut down again.  You can try to diagnose exactly what the cause is, but often it is easier to just replace both major components.

Out here in L.A. I think the water has something in it that affects the plastic shut-off valves and causes them to stop working.  I think there was a period where I had to replace one once a year in my apartment over the course of ten years.  Finally, after trying all the different brands and types, I got one that was durable and foolproof.  Most filler valves are controlled by a float, and are adjustable for fill level in some way.  There are two basic types as seen in this picture.