Just before we started moving into our new condo, I noticed that our toilet was continually running, and that unfortunately, it was a fairly fast leak. Step one, was to determine the source of the leak. Our toilet has a standard flapper-flush-valve, so the leak was almost certainly with the flapper.
Incidentally, we have really hard water in our area, which is why the toilet tank looks so filthy. It's vaguely metallic smelling, and it'd be a bitch to clean, but the inside of the tank isn't such a concern for me.
After turning off the water and draining the tank, the leak stopped immediately, ensuring that, sure enough, the flapper wasn't sealing properly anymore. Excellent, because the alternative was that the tail piece was leaking, and that would be a lot more of a pain in the ass to fix. Plus, the flapper is a sub-$5 part, depending on which one you buy. I opted for one from a brand named 'Corky' which was about $4, but they're all pretty much the same.
Actually, the only difference between assemblies is whether the flapper needs a ring to slip all the way down the overflow pipe, or if it attaches via some small clips on a plastic ring already built on to the flapper. Most flappers on the market today, are designed to be used in either system, plus, they tend to be bigger than the one that was on our toilet.
Our toilet had the clips on the side, so the ring has to go, but luckily, a utility knife is enough to remove the ring and the back piece and install the new flapper, which will just hook directly on the same attach points as the old flapper. Turn the water back on, you might need to flush once to ensure a good seal between the flapper, but it's a pretty simple, painless fix, something that doing yourself will cost $5 and fifteen minutes, but would cost probably $60 for a plumber to come out and fix (based on estimates of plumber time).