The impossible bookkeeping problem
A litre of air at room temperature holds about 2.5\times10^{22} molecules. In principle Newton's laws determine every collision — but writing 10^{22} coupled equations, and knowing 10^{22} initial positions and velocities, is not merely hard, it is meaningless. No computer could store the data; no measurement could ever supply it. And yet the gas obeys the ideal gas law PV=Nk_BT to exquisite precision, every time.
This is the miracle statistical mechanics explains: how the lawless motion of countless particles produces a handful of lawful, reproducible quantities — pressure, temperature, entropy. The trick is to stop tracking individuals and start counting possibilities. We give up predicting where any one molecule is, and gain the ability to predict, with near-certainty, what the whole ensemble does.
Microstate versus macrostate
The single most important distinction in the subject. A microstate is a complete specification of the system at the atomic level — every position and momentum (classically), or every quantum number (quantum-mechanically). A macrostate is what you can actually measure: the total energy E, the volume V, the number of particles N. Enormously many microstates share the same macrostate.
Concrete picture: flip four coins. The microstate is the exact face of each coin, HTHT say — there are 2^4=16 of them, all equally likely. The macrostate is just the number of heads. The macrostate '2 heads' is realized by 6 different microstates (HHTT, HTHT, …), while '4 heads' is realized by only 1. The macrostate you are overwhelmingly likely to observe is simply the one that the most microstates produce.
The fundamental postulate
Statistical mechanics rests on a single, deceptively simple assumption, the fundamental postulate: for an isolated system in equilibrium, every accessible microstate of a given energy is equally probable. There is no reason to prefer one arrangement of the atoms over another with the same energy, so we prefer none. Everything else — entropy, temperature, the whole edifice — follows from this.
The postulate cannot be proved from mechanics alone; it is justified because the predictions it yields agree with every experiment, and because chaotic microscopic dynamics does, in practice, sweep a system through its accessible states roughly uniformly over time (the ergodic idea). Treat it as physics' most successful working hypothesis.
Entropy is counting
Let \Omega(E,V,N) — the multiplicity — be the number of microstates consistent with a given macrostate. Boltzmann's monumental insight, carved on his tombstone, is that entropy is just the logarithm of this count:
The Boltzmann entropy formula. k_B is the Boltzmann constant. The logarithm makes entropy additive: combine two independent systems and their multiplicities multiply (\Omega=\Omega_1\Omega_2), so their entropies add (S=S_1+S_2).
This is the bridge, the Boltzmann entropy formula. On the left, a thermodynamic quantity measured with heat and thermometers. On the right, a pure count of atomic arrangements. The mysterious 'entropy' of Vol I is revealed to be a measure of how many ways the microscopic world can produce what you see. Higher entropy simply means more microstates — more ways to be that macrostate.
Why the biggest macrostate always wins
The second law of thermodynamics now looks almost obvious. A system left alone explores its accessible microstates; since all are equally likely, it spends essentially all its time in whichever macrostate owns the most of them — the maximum-\Omega, maximum-entropy macrostate. Equilibrium is not a special arrangement; it is simply the most probable one, and with N\sim10^{23} it is more probable than the alternatives by unimaginable factors.
How overwhelming? For a gas of N molecules the probability of a macroscopic fluctuation away from equilibrium scales like e^{-N}. With N=10^{23}, spontaneously finding all the air in half the room is not 'rare' — its probability is e^{-10^{23}}, a number so close to zero that it would not happen once in a trillion ages of the universe. This is why thermodynamic laws feel absolute even though they are 'only' statistical.