You could replace all of the insides of the tank - but you could save time and money by first figuring out what the problem is.
*It could be that the flapper has gone bad and needs to be replaced. That'll cost ya about $3. If you hear a low-pitched water running sound (the same sound you hear when you just barely hold the flush handle down) then start by replacing the big rubber flapper. (It's easy - go the website that (Henry was it?) recommended.
*On the other hand - it could be the adjustment of the valve. Take off the top of the tank (And if you DO drop it, try not to drop it inside the toilet. Learned the hard way how much damage that can cause.) Pull up on the ball float (big ball floating in the tank.) If the water stops running, the float just needs to be adjusted (down.) There are different types of toilet apparatus, so the adjustment may be done one of several ways. If the ball is at the end of a metal rod, you can just bend the rod down. If it's at the end of a plastic rod, it probably has a little plastic wing-nut/screw that you can loosen, adjust down the float, and re-tighten.
If these don't fix it, you may need to replace the valve. However, if you have to replace parts and you haven't yet tried the two things I suggested, you may have trouble installing & adjusting the new parts.