For being such a simple contraption, it's amazing how pesky these things can be.  Luckily, the two functions (flushing and re-filling) are isolated and are usually fairly easy to diagnose and fix. 

Toilet Tip:  I know it sounds counterintuitive, but if you've turned off the water supply and want to empty the bowl, quickly dump a great big bucket of water into it.  Somehow, the weight of all the water forces it en masse past the trap part and the bowl should be fairly empty. 
  

RUNNING 

This can be either the flapper or the valve.  A quick inspection will pop up the culprit: 

Lift off the lid and look at the overflow pipe.  If water is running into it, then the valve either needs adjustment or replacing.  If it's the old style with the big long arm and 'bulb' on the end, either turn the adjustment screw or just give the arm a slight bend so it'll shut off the water sooner.  If it's the newer 'single-unit' type, lower the little clip on the side.  If the valve feels like it's shut off but water is still getting through, replace it.  If you want to pop the top off to see if something needs cleaning, turn off the water first.

  

If water isn't going into the overflow tube, it's probably the flapper allowing water to leak past it.  They're cheap and easy to replace.  Just make sure you bring the old one with you to the store, as there are a handful of types out there.

  
If the toilet runs very seldomly, the flapper is probably leaking just a tiny bit and might just need a good cleaning.  Take an abrasive pad, like an SOS pad, and clean both the underside of the flapper around the edge and the plastic ring it presses against. 
 

LEAKING ONTO FLOOR

This will most likely be one of two different seals leaking, either the one between the tank and the bowl, or the one between the bowl and the floor.

Neither job is particularly tough, but it depends on how rusty the bolts are.  For the tank seal, turn off the water and unhook the line, then unscrew the bolts holding the tank to the bowl.  Replace the 'doughnut' and you're good to go.  Just make sure you re-tighten the bolts the same amount so the tank stays level, and just squeeze them down until everything's nice and snug.  Porcelain snaps pretty easily and demands respect.

For the floor seal, turn off the water, unhook the supply line, then undo the two bolts holding it to the floor.  Maybe with some help, lift the toilet out of the way.  Scrape off the old wax, put the new seal in place, then carefully sit the toilet down on the new seal.  Don't overtighten the floor bolts.
 

REPLACEMENT

Basically, the same as replacing the floor seal.  Get the old toilet unhooked, clean up the old seal, put down a new one then assemble the new toilet.  These things are usually standardized, so hopefully the floor bolts and supply line will fit okay.

One thing that might be surprising is that there's no foul odor coming from the hole in the floor.  Sewage systems have a slight 'draw' on them, praise be.