Counterfeit Masses

So, a recent discussion with Simon caused me to remember an old puzzle I've been meaning to post for some time. There are multiple versions of this puzzle though, so I'm going to start off with the easiest one and move up. I don't know where I first learned this puzzle, I've known it since childhood:

You have a balance scale and 27 coins. 26 of the coins are identical while the other one is a counterfeit. The only difference with a counterfeit coin is that it is slightly heavier than other coins. You may only use the balance scale to make 3 measurements, find an algorithm that is guaranteed to identify the counterfeit coin.

Of course, the amount by which the counterfeit coin is heavier is imperceptible to a human, only a balance scale can tell. Using the balance scale to make a measurement can be visualized as follows: The scale has two places to put coins, you put as many coins as you like in each of those two places, then you push a button and the scale tells you which of the two places has more mass on it. The button will work 3 times and then the scale will break.