Leptogenesis is a process, usually taking place in the early Universe, that generates a difference between the number of leptons and anti-leptons.This lepton asymmetry can contribute to the matter excess observed in the Universe.
After inflation, the Universe is believed to have been "reborn" containing equal amounts of matter and anti-matter. Nonetheless, the visible Universe today appears to be made of matter — and not anti-matter. Since matter is made of atoms, and atoms are made of electrons orbiting neutrons and protons (both a type of baryon), this matter asymmetry is referred to as the Baryon Asymmetry of the Universe. (The Lepton Asymmetry of the Universe is not measured, because as well as the electrons in atoms, the Universe contains enormous but poorly measured populations of neutrinos and anti-neutrinos. The neutrinos are leptons, so a large asymmetry could be hidden among them.) To generate the baryon asymmetry after inflation, requires three ingredients (originally given by Sakharov). Firstly, the matter and anti-matter must behave differently. This is refered to as C and CP violation (because C and CP are transformations that interchange particles and anti-particles). Secondly, some interactions must violate baryon number. This can be delicate to implement, because baryon number violation is constrained by the measured lower bound on the proton lifetime. Finally, the generation of the excess is a dynamical process, that is, the system responsable for the excess cannot be in equilibrium.
There are many realisations of the leptogenesis mechanism, such as "Affleck-Dine" in supersymmetric models, out-of-equilibrium decay in seesaw models (this includes thermal leptogenesis), etc. They differ in their sources of CP violation and non-equilibrium. The common characteristic is that a lepton asymmetry is generated, then partially reprocessed into a baryon asymmetry by non-perturbative baryon + lepton number violating Standard Model processes, which are refered to as sphalerons. These transform three anti- leptons (one from each generation), into 9 quarks (of each generation and colour). Although these non-perturbative processes are predicted by the Standard Model, for reasons related to anomalies, and are expected to be fast in the early Universe, they are exponentially suppressed today. So they are consistent with our observation that baryon and lepton number are separately conserved today.