Wikipedia is your friend for this question, but for starters one can say that Green's functions are solutions for certain well defined differential equations.
Knowing the Green function allows you to solve inhomogeneous linear PDEs for pretty much any nonhomogeneity (i.e. right-hand side). This mathematical fact has many physical implications not just in electrodynamics (causality etc. -- see propagators), cf.e.g. the links below
As Artur Sergyeyev noticed, Green's function allows you to find the solutions of inhomogeneous differential equation. You can learn this formalism reading chapter 2.10, p. 25 in "Principles of Nano-Optics" written by L. Novotny and B. Hecht. Here you can find derivation of the Green's function for electric field and understand certain applications.
The Green function- tensor denotes mathematically the field generated by a point charge- or current. Its importance stems from this and the linearity of the Maxwell equations, or many other propagator equations:
The field generated by a source can be thought as being the superpositions of the fields generated by each of its constituting point sources.