a gauge transformation
The potentials V and A are not unique — many different pairs give exactly the same physical fields E and B. A gauge transformation is a coordinated re-labelling of the potentials that changes them while leaving every measurable field untouched. This freedom, at first an annoyance, turns out to be one of the deepest ideas in physics.
Pick any scalar function lambda(r, t). The gauge transformation replaces A by A + grad lambda and simultaneously replaces V by V - dlambda/dt. Check the fields: B = curl A is unchanged because the curl of a gradient is zero, and E = -grad V - dA/dt is unchanged because the two new pieces cancel. So E and B — the things you can measure — are gauge-invariant, while V and A carry a redundancy you are free to fix however is convenient.
This is why gauge choices like the Coulomb gauge (div A = 0) or the Lorenz gauge (div A + (1/c^2) dV/dt = 0) are allowed: they pin down the leftover freedom without touching the physics. Promoted to a local symmetry, this same gauge principle — demand invariance under position-dependent phase changes and a gauge field must appear to compensate — generates electromagnetism itself, and its non-abelian generalizations generate the weak and strong forces of the Standard Model.
Add lambda = C t (a constant times time) to the gauge: A stays put (grad lambda = 0) but V shifts to V - C everywhere. The potential changes by a constant, yet E = -grad V is untouched because the gradient of a constant is zero. Two different V's, one identical field — the hallmark of gauge freedom.
A goes to A + grad lambda, V goes to V - dlambda/dt leaves both E and B unchanged.
Only gauge-invariant quantities are physical; any result that depends on your choice of gauge is a mistake. Gauge freedom is a redundancy in the DESCRIPTION, not a symmetry that moves the system to a new state — nothing observable changes.