Definition
Entropy is a quantitative measure of disorder, or equivalently the logarithm of the number of microscopic configurations (microstates) consistent with a given macroscopic state.
In Boltzmann's formula S = k log W, k is Boltzmann's constant and W is the count of microstates. A system with more equivalent ways to be arranged has higher entropy.
Why it matters
How it works
The thermodynamic definition treats entropy as the quantity dS = dQ_rev / T — heat added reversibly, divided by temperature. This was Clausius's nineteenth-century formulation, devised before the atomic theory was settled. It explains why heat flows spontaneously from hot to cold (entropy increases) but never the reverse (entropy would decrease).
Ludwig Boltzmann supplied the microscopic interpretation. A gas of N molecules has astronomically many microstates — specific positions and momenta — consistent with the macroscopic temperature and pressure. Equilibrium is overwhelmingly the most probable macrostate simply because it corresponds to the largest number of microstates. The system tends to higher entropy not because of any new force, but because the disordered configurations vastly outnumber the ordered ones.
Claude Shannon later showed in 1948 that the same mathematical structure describes information: the entropy of a message source measures the average number of bits needed to encode it. Connecting thermodynamic entropy to information-theoretic entropy is what underlies modern arguments about Maxwell's demon, the Landauer limit on irreversible computing, and — crucially for Hawking's work — the information carried by black holes and their radiation.