Many problems that students solve in introductory physics classes have to do with one or two interacting bodies. The two-body problem, in which two objects interact with a force dependent on the distance between them, is often exactly solvable. Take the Earth and the Moon, or the Earth and the Sun, and you can exactly solve the simple equations of motion by simplifying the two-body system into a one-body system. The three-body problem, on the other hand, is intractable. Add Jupiter or another planet into the bargain and suddenly all the math in the world won't help you except to numerically approximate the system's behavior. There are all sorts of approximations that can help you out, but the exact solution for arbitrary conditions is elusive. This, of course, makes analysis of complicated systems...well, complicated. Luckily for physicists, while the 3-body, 4-body, and 5-body systems are difficult or impossible mathematically, the $10^{23}$-body system is actually solvable in a statistical sense. The examination of systems with many many constituent parts (atoms, for instance) forms the basis of statistical mechanics, an incredibly powerful framework that describes large systems and explains tons of other physics into the bargain.
Statistical mechanics is based on just two simple hypotheses. As far as I know, these two hypotheses aren't provable from more basic principles, but taking them as axioms gives all sorts of useful and accurate results. In order to discuss these hypotheses, I first need to define a couple of terms. Any large system can be described in terms of a (very large) collection of microparameters. These do things like saying exactly where each particle is and what its momentum is, or the energy level of each particle, and so on. The exact microparameters will depend on the system we're examining. In general, it's very difficult (not to mention time-consuming) to measure microparameters. Instead, we define macroparameters to describe the system as a whole. Things like temperature, pressure, or the total number of spin-up atoms are all macroparameters, and again, we tend to define different macroparameters for different systems.
With those definitions out of the way, let's take a look at the first hypothesis of statistical mechanics: the ergodicity hypothesis. It states that for any system, there is an equilibrium state, in which the macroparameters describing the system (no matter which ones we choose) are constant, and that furthermore, the values of these macroparameters are given by the unweighted statistical average of all the microstates of the system.
This is a very profound statement, on a variety of levels. The existence of an equilibrium state for any old system is moderately intuitive, but it seems a little strange to have such an overarching statement of its existence. More than that, though, it gives an extraordinarily helpful way to find the equilibrium values of all the macroparameters: just take the average of all the microstates! Naturally, the math behind such averages is a little messy at times, but it's of critical importance that we can so easily find a description of equilibrium.
Alright, time for one more definition: let's call any system in which the microstates change faster than macroscopic time scales an ergodic system. In order for a system to experience this flavor of fast evolution, it is necessary to have interactions between particles in the system - imagine taking a collection of noninteracting gas particles and squishing them into a small ball at the center of a room; in the absence of interactions, nothing will ever happen and the system won't evolve at all.
The second critical hypothesis of statistical mechanics is called the relaxation hypothesis, and it states that any ergodic system always evolves towards equilibrium. This translates roughly into the third law of thermodynamics, which states that entropy always increases.
Based on these, we can derive (with sufficient approximations) things like the ideal gas law, rigorous definitions of temperature and pressure, and in general the behavior of very large systems.
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