The thermodynamics potentials for describing matter at nonzero temperatures and densities or chemical potentials are summarized. Emphasis is put on the thermodynamically correct description within the canonical and grand canonical ensemble for dense matter. The notion of chemical equilibrium is introduced for several conserved quantities and used to describe matter in β-equilibrium where charge and baryon number are conserved. The limit for nonrelativistic and relativistic particles is worked out in detail. The concept of an equation of state is introduced and applied to free Fermi gases. The pressure integral is solved analytically and the nonrelativistic and relativistic limits for the equation of state are delineated. Finally, the properties of polytropes are discussed and connected to the limiting cases of the equation of state of a free Fermi gas.