the four-potential
In ordinary electromagnetism you meet two potentials — the scalar potential V and the vector potential A. Relativity reveals they are not two objects but one: the four-potential, a single four-vector whose time part is V and whose space part is A.
The four-potential is A^mu = (V/c, A_x, A_y, A_z), transforming as a four-vector under Lorentz boosts. From it the entire electromagnetic field follows, F^(mu nu) = d^mu A^nu - d^nu A^mu. The old gauge freedom — that E and B are unchanged if you shift V and A together in a coordinated way — becomes the crisp statement A^mu goes to A^mu + d^mu lambda for any scalar function lambda. In the Lorenz gauge, defined covariantly by d_mu A^mu = 0, Maxwell's equations reduce to four uncoupled wave equations box A^mu = mu_0 J^mu, where box is the d'Alembertian — one clean relation replacing the tangle of vector-calculus equations.
The four-potential is more than bookkeeping. It is the field that couples to charge in the Lagrangian, the object promoted to an operator in quantum electrodynamics, and — through the Aharonov-Bohm effect — the carrier of physically real information even where E and B both vanish. Its gauge symmetry, d_mu A^mu structure and all, is the template for the gauge theories of the entire Standard Model.
For a static point charge the four-potential is simply A^mu = (V/c, 0, 0, 0) with V = q/(4 pi epsilon_0 r) — purely timelike, no vector potential. Boost to a moving frame and the time and space parts mix: the same A^mu acquires a nonzero A, and a magnetic field appears, just as a moving charge should have.
A^mu = (V/c, A) unifies the scalar and vector potentials; in Lorenz gauge box A^mu = mu_0 J^mu.
The four-potential is not unique — the gauge transformation A^mu goes to A^mu + d^mu lambda leaves the fields F^(mu nu) unchanged, so it is A^mu that carries the redundancy, not the physics. Only gauge-invariant quantities (the fields, and loop integrals of A) are observable.