Chemical potential, μ, was defined by Josiah Willard Gibbs as:
If to any homogeneous mass in a state of hydrostatic stress we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered.
A feel for this potential:
Being a potential μ is of the same character as a pressure or a voltage or even a temperature. It sort of "pushes" on things.
If a pressure is allowed to alter the volume of a system by a tiny amount then the internal energy, U, (the total energy of the system) changes by the small amount of work that has been added to the system. Equally if temperature is allowed to push the thing through an entropy change heat energy increments the internal energy.
μ may push material into the system, perhaps because of a concentration gradient or electrical repulsion between particles.
This is seen in the basic identity of thermodynamics, the thermodynamic identity: dU = T.dS - P.dV + μ.dN (U internal energy, T temperature, S entropy, P pressure, V volume, μ chemical potential) Where only one type of particle is involved for simplicity - for more than one simply sum the separate terms.