In thermodynamics, H-theorem is a Newtonian mechanics based equation that quantifies the heat of an ideal gas, by a numerical quantity ‘H’, short for heat, defined in terms of the velocity distributions of the atoms and molecules of the body of gas, such that when the system particles fall into a Maxwell-Boltzmann distribution (Gaussian distribution), H assumes it minimum possible value and that any gas system not at its minimal value of H will transform its distribution of velocities, through collisions, towards the minimum value associated with the Maxwell-Boltzmann distribution.  In simple terms, the H-theorem is kinetic theory-based definition of entropy, albeit applicable to only gas systems, and is often claimed as an analytical proof of the second law of thermodynamics. 

History
In 1866, Irish physicist James Maxwell published his “On the Dynamical Theory of Gases”, showing how, through kinetic theory, to obtain physical properties of gases from the underlying distribution of velocities. Building on this work, Austrian physicist Ludwig Boltzmann developed what he called a ‘minimum theorem’, in his famous 1872 paper “Further Studies on the Thermal Equilibrium of Gas Molecules”, which derived a kinetic expression for the entropy of an ideal gas.

The term ‘H-theorem’ was said to have been an English translation of Boltzmann’s term ‘minimum theorem’, made by English mathematical physicist Samuel Burbury between 1875 and 1894, in the sense of Boltzmann's 'heat theorem'. In his 1899 book A Treatise on the Kinetic Theory of Gases, which was essentially devoted to an explication of Boltzmann’s 1872 work, Burbury states that the H theorem is known as ‘Boltzmann’s minimum theorem’ and that H is called ‘Boltzmann’s minimum function or, as it is sometimes, called the entropy function.’ 

References
1. Lindley, David. (2001). Boltzmann’s Atom: the Great Debate that Launched a Revolution in Physics (H, pg. 75, Burbury, pgs. 109-22). Free Press.
2. Burbury, S. H. (1876). “On the Second Law of Thermodynamics, in Connection with the Kinetic Theory of Gasses”, Philosophical Magazine, 1: 61-67.
3. Burbury, S. H. (1899). A Treatise on the Kinetic Theory of Gases (H theorem, pgs. v-vii, 10, 30, 39-45, etc.). Cambridge University Press.