a thermodynamic potential
In mechanics a single potential energy function, once known, tells you the forces and the equilibria — you just take derivatives. Thermodynamics has the same luxury, but with a twist: which 'master function' you use depends on what your experiment holds fixed. A thermodynamic potential is a state function engineered so that its minimum picks out equilibrium under a specific set of constraints, and so that every other thermodynamic quantity falls out by differentiation.
The standard family is the internal energy U(S,V,N), the enthalpy H(S,p,N), the Helmholtz free energy F(T,V,N), the Gibbs free energy G(T,p,N) and the grand potential Omega(T,V,mu). Each is written in its own natural (or 'characteristic') variables, which come in conjugate pairs — temperature with entropy (T,S), pressure with volume (p,V), chemical potential with number (mu,N). Neighbouring potentials are linked by Legendre transforms, e.g. F = U - T S swaps S for T. The second law becomes: at fixed values of a potential's natural variables, that potential is minimized in equilibrium.
Choosing the right potential is the practical art of thermodynamics: isolated system -> maximize entropy; constant T and V -> minimize F; constant T and p -> minimize G; open at fixed mu -> minimize Omega. Crucial caveat: a potential carries complete information only in its natural variables. Writing U as a function of T and V instead of S and V throws information away — you can no longer recover everything by differentiation.
A gas in a rigid, insulated box (fixed U, V, N) maximizes its entropy at equilibrium. Move it to a thermostat at temperature T (fixed T, V, N) and the relevant statement flips to: it minimizes the Helmholtz free energy F = U - T S. Same gas, different constraint, different potential doing the work.
The constraint you impose selects which potential is extremal at equilibrium.
A thermodynamic potential is a 'characteristic function' only in its natural variables; expressed in any other variables it still describes the system but is no longer a complete generating function for all its properties.