Poisson's equation (Poisson equation) is an equation template that fits a number of physical phenomena, including those governed by an inverse square law, such as gravitational fields and electric fields. It asserts a particular relationship between two mathematical fields, showing that each of them models a particular physical field in a different manner:
∇²f = g or ∇·∇f = g or Δf = g
Through such an equation, given either the above f or g and some boundary conditions, the other can be determined. The function f is termed a potential (e.g., gravitational potential), ∇f (gradient of f) is a vector field indicating the force-per-whatever (for gravity, the force-per-mass, equivalent to the resulting acceleration, given f=ma) and g is the (gravitational) mass, or (for the electric field example) the charge. The vector field ∇f has a zero curl, as does any gradient field. Multiple masses or charges can be modeled by summing the two sides of the corresponding Poisson equations associated with each mass/charge.
Laplace's equation amounts to a special case of Poisson's equation, when function g is zero, i.e., evaluates to zero for all parameters:
∇²f = 0
It is of note that for fields f and g described by Poisson's equation, throughout any volume in which g happens to be zero, Laplace's equation is holding over that volume. If Poisson's equation is used to model the field surrounding a single point (or single spherically symmetric) charge or mass, then any region beyond the charge/mass is adhering to Laplace's equation.