Wikipedia
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, (often simply called the vector potential) and the magnetic scalar potential, ψ. Both quantities can be used in certain circumstances to calculate the magnetic field.
The more frequently used magnetic vector potential, A, is defined such that the curl of A is the magnetic field B. Together with the electric potential, the magnetic vector potential can be used to specify the electric field, E as well. Therefore, many equations of electromagnetism can be written either in terms of the E and B, or in terms of the magnetic vector potential and electric potential. In more advanced theories such as quantum mechanics, most equations use the potentials and not the E and B fields.
The magnetic scalar potential ψ is sometimes used to specify the magnetic H-field in cases when there are no free currents, in a manner analogous to using the electric potential to determine the electric field in electrostatics. One important use of ψ is to determine the magnetic field due to permanent magnets when their magnetization is known. With some care the scalar potential can be extended to include free currents as well.
Historically, Lord Kelvin first introduced the concept of magnetic vector potential in 1851. He also showed the formula relating magnetic vector potential and magnetic field.