Definition
The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and stays constant only in the idealized limit of a reversible process.
Equivalent statements: heat does not flow spontaneously from a cold body to a hot one (Clausius); no engine can convert heat entirely into work with no other effect (Kelvin-Planck); the universe tends toward thermodynamic equilibrium.
Why it matters
How it works
Sadi Carnot's 1824 analysis of heat engines showed that no engine running between a hot and a cold reservoir can be more efficient than a reversible (Carnot) engine — and that this efficiency depends only on the temperatures. Clausius generalized the result by defining entropy and observing that in any real process the entropy of the universe increases. Kelvin and Planck reformulated the same content as the impossibility of perfect heat-to-work conversion.
Boltzmann supplied the microscopic explanation. The macrostate of a system corresponds to vast numbers of microstates, and overwhelmingly most microstates are near equilibrium. A non-equilibrium state — gas concentrated in one corner of a room, for instance — has very few microstates compared with the uniformly-distributed state. Time evolution explores the available microstates more or less uniformly, so a typical trajectory wanders toward the macrostate with the most. Reversals are not forbidden, they are stupendously unlikely: the probability of a room's worth of air spontaneously concentrating in one corner is essentially zero on any timescale relevant to physics.
The law took on new significance when Bekenstein and Hawking showed black holes obey a "generalized second law": S_total = S_matter + A/4 (in Planck units) never decreases. Throwing a hot cup of tea into a black hole reduces ordinary entropy outside, but the resulting increase in horizon area more than compensates. The second law thus extends from thermodynamics into general relativity, and ultimately into quantum gravity — one of the few principles believed to survive in every regime.