A vector field is basically a vector quantity defined at every point in some space - imagine it as a continuum of arrows drawn flowing through space. Some vector fields have the property of being conservative, such as the electrostatic field; this means that the arrows can be thought of in some sense as the gradient of some surface (not true for all fields, imagine one in which the arrows go around in a circle).
Formally, given such a field F, we say that there exists a scalar function ρ such that F = gradρ,and we say that ρ is a scalar potential.