In particle physics distribution functions describe the energies of particles in a system depending on its temperature. Even in systems which appear to be at a single temperature, such as the particles in air, some will be moving faster or slower than others. Understanding the distribution function is important for evaluating microscopic and macroscopic properties of a thermodynamic system.
Three of the most important examples of distribution function are the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions; the former is a classical distribution, appropriate for identical but distinguishable particles, and the latter two are quantum distributions, used for fermions and bosons respectively.