Given a linear differential operator (ordinary or partial) $P$ on a domain $M\subset\mathbb{R}^n$ or a manifold $M$, one can consider both the homogeneous equation $P f = 0$ and the nonhomogeneous equation of the form $P f = g$ where $g$ is a given nonhomogeneous term. If $g$ is a delta function and the boundary conditions are given, then the solution of the nonhomogenous equation
is called the fundamental solution for $P$; alternative names like Green function and function of influence are also used. A particular solution of the nonhomogeneous equation for some other $g$ can be obtained by calculating the convolution with the fundamental solution. (Compare the fact that the delta function is the identity element for convolution.)
Wikipedia, Fundamental solution
Wikipedia, Green’s functions