If two bodies are each in thermal equilibrium with a third body, they are in thermal equilibrium with each other.

In slightly plainer english: If my can of Coke is the same temperature as Jeff's can of Dr Pepper, and Nicole's beer is the same temperature as my Coke, then Nik's beer is the same temperature as Jeff's Dr Pepper. And both of us are wondering what Nik is doing drinking beer at work.

Upon first reading this, one might tend to react with the following phrase: "Well, Duh!" This is, after all, a physical analog to the transitive property of equality.

Of course, we know this to be true inherently from our everyday experience. However, long after The Three Laws of Thermodynamics had been accepted and gone into general use in physics and chemistry, it became clear that if this weren't true, the Three Laws wouldn't be valid. Physicists are in the business of making sure that our fundamental beliefs are not in fact a load of dingos' kidneys, so we decided we needed to make sure we dotted our i's and crossed our t's on this one.

It does occur to me, though, that this may really be an embodiment of The First Law of Thermodynamics, namely, the law of conservation of energy (after Einstein, conservation of Mass/Energy). My argument follows thusly:

If Objects A & C are in thermal equilibrium with object B, but not with each other, then a heat engine can be made to extract power from the temperature difference. However, by doing so, you've taken A & C out of equilibrium with B. You can now use a heat engine to bring A & B into equilibrium. However, now both are out of equilibrium wiht C. Et Cetera. I haven't done the math on this to work out if that can be an infinite loop, but it certainly demonstrates why it is that the Zeroth Law is a fundamental concept in Thermodynamics.

*thanks to mblase for clueing me in to the transitive property bit.