the scalar potential
Rather than track the three components of the electric field everywhere, it is often far easier to track a single number at each point and take its slope. The scalar potential V is that number — the electrostatic potential energy per unit charge — and in statics the entire electric field is just its downhill gradient.
In electrostatics, because curl E = 0, the field can be written as the gradient of a potential, E = -grad V; the minus sign makes E point from high to low potential, downhill. Feeding this into Gauss's law gives Poisson's equation laplacian V = -rho/epsilon_0, which reduces to Laplace's equation laplacian V = 0 wherever there is no charge — one scalar equation replacing three vector ones. When fields change in time the story enlarges: Faraday's law forces curl E to be nonzero, so E = -grad V - dA/dt, and the scalar potential shares the job with the vector potential A.
The potential is not unique. You can add any constant to V without changing E — only potential differences are physical — and in the full time-dependent theory you can trade freedom between V and A through a gauge transformation. What is directly measurable is the field E, or equivalently the work per charge V(b) - V(a) between two points; the absolute value of V is a convention, which is why we are free to set V = 0 at infinity or at ground.
A point charge q sets up a scalar potential V = q/(4 pi epsilon_0 r), a single number falling off as 1/r. Its gradient, E = -grad V = q/(4 pi epsilon_0 r^2) pointing radially outward, recovers Coulomb's field — one scalar function encoding the whole vector field.
E = -grad V in statics; via Gauss's law this becomes Poisson's equation laplacian V = -rho/epsilon_0.
E = -grad V holds only in electrostatics; once magnetic fields change in time you must include the vector-potential term, E = -grad V - dA/dt. The zero of potential is arbitrary — only differences of V, not its absolute value, are physical.