A toilet that starts running at random times, runs for a few seconds and then quits, is leaking water from the tank. If there is no water on the floor the tank is leaking into the bowl. This is usually repaired by replacing the flapper, or less often the toilet tank gasket. The flapper itself is inexpensive and can be easily replaced by a home owner. The toilet tank gasket requires a little more care to install as you can easily over tighten the bolts and break the porcelain tank.
A randomly running toilet as well as water on the floor means the tank is leaking externally. The toilet tank gasket could again be the culprit, or the supply line connection could be leaking. Fixing either should solve the problem.
Water on the floor with out the random running could be one of several problems. A supply line leak is an easy fix for the home owner. If the supply stop itself is leaking it would be best for a plumber to make repairs. The wax ring under the toilet could also need replacing. Since this task requires actually removing and reinstalling the toilet, most home owners prefer not to tackle it themselves.
Whistling or whining noises during the flush-refill cycle will usually indicate a problem with your fill valve. This can sometimes be as simple as cleaning trash out of the fill valve assembly and reassemble. Otherwise, the entire flush valve can usually be replaced with a new one.
A flush valve that isn't getting enough water can also make unusual noises. Make sure the supply stop is turned fully on.
The most common flushing problem is a broken/disconnected chain that connects the flush handle lever to the flapper. This is a quick and easy fix. Make sure that there isn't too much slack as that will keep the flapper from opening fully. Not enough slack will cause the same problem. Adjust the chains length until the toilet flushes properly.
A complete flush requires proper water level in the tank. If the water level is too low or too high it can be changed by adjusting the float level. Be careful not to set the water level too high as it will not help the toilet to flush better. It will only force the toilet to flush longer and waste water.