What does the zeroth law of thermodynamics define? Thermal equilibrium or temperature? If zeroth law defines temperature what defines thermal equilibrium? Do we define thermal equilibrium first and then go on defining the temperature? Thanks!
 A: It defines thermal equilibrium, or more precisely it identifies thermal equilibrium as a transitive relation. 
In principle there are a bunch of different temperature scales which are functions of our "absolute" positive scale and come from various measurement apparatuses that we might bring into thermal equilibrium with an object and then they give us back some number. Like, we might appreciate if those scales are monotonically increasing so that we never have heat flow from a "colder" body to a "hotter" body according to that scale, but that doesn't distinguish $T$ from $\sqrt{T}$ or $T^2$. In fact there's a decent argument that our "absolute" scale is actually $T = 1/C$ of some more fundamental "coldness" scale, which allows us to better understand what "negative temperatures" are (negative-absolute-$T$ temperatures turn out to be hotter than positive-$T$ temperatures, a fact which is more obvious when looking at coldness $C$).
Our absolute scale depends on the idea of a small-and-long ideal piston storing an ideal gas: changes in its $T$ then show up in the displacement of the piston length $x$ from what it would be if there were no gas in there at all, allowing us to measure temperature; the fact that it is small just means that it does not disturb the other object's temperature much when we take the measurement.
But the fact that this is a transitive relation is more primitive. It's what makes thermometers possible in the first place. Thermometers are only useful because when I measure two temperatures as the same, I know that if I bring those two into thermal contact, they won't exchange heat. Imagine if I were measuring the wrong thing, for example wetness: I could measure your hand as some temperature $T$ ("dry") and measure the stove at the same temperature $T$ (equally "dry") but placing your hand on the stove could still burn you: then this temperature measurement fails to tell you what you really wanted to know. If the zeroth law weren't there then we would be admitting that it's impossible to tell you what you really wanted to know. Thankfully, it's not.
