George Green was a British mathematician and physicist, who wrote the long-lived monograph: An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Green, 1828).

The essay introduced several important concepts, among them a theorem similar to modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions. These notions, Green's functions and Green's theorem have been widely used and applied.

Green was the first person to create a mathematical theory of electricity and magnetism and his theory formed the foundation for the work of other scientists such as James Clerk Maxwell, William Thomson, and others. His work on potential theory were parallel in part to that of C. F. Gauss.
In short, George Green transformed the differential equations of electromagnetic problem into integral equations by means of kernels that have attained the generic name Green functions. An original one is the Coulomb potential, the Green function of the Poisson equation.
Scattering theory, particularly in the Born type approximation, employs Green functions as well as many-body theory has shown their general versatility. It is most often the case that the argument starts from a time-dependent formulation and that a subsequent Fourier transformation generates the energy representation, the spectral forms, and analytical features.

Applications of the Green's functions method in the many-body theory and comprehensive list of References are given in the works:
A. L. Kuzemsky, Irreducible Green Functions Method and Many-Particle Interacting Systems on a Lattice.
Rivista del Nuovo Cimento, vol.25, p.1 (2002) [e-Preprint: arXiv:cond-mat/0208219].

A. L. Kuzemsky, Statistical mechanics and the physics of many-particle model systems.
Physics of Particles and Nuclei, vol.40, p.949 (2009) [e-Preprint: arXiv:1101.3423v1 [cond-mat.str-el] 18 Jan 2011].

The Green function technique is a method to solve a nonhomogeneous differential equation. The essence of the method consists in finding of an integral operator which produces a solution satisfying all given boundary conditions. The Green function is the kernel of the integral operator inverse to the differential operator generated by the given differential equation and the homogeneous boundary conditions. It reduces the study of the properties of the differential operator to the study of similar properties of the corresponding integral operator.
The integral operator has a kernel called the Green function, usually denoted G(x,t). This is multiplied by the nonhomogeneous term and integrated by one of the variables.
In other words, the Green functions play an important role in the effective and compact solution of linear ordinary and partial differential equations. They may be considered also as an crucial approach to the development of boundary integral equation methods.

Green was born and lived for most of his life in the English town of Sneinton, Nottinghamshire, now part of the city of Nottingham. His father, also named George, was a baker who had built and owned a brick windmill used to grind grain. Green's work was not well known in the mathematical community during his lifetime.
There are a few good sources on his biography and works:

Yu. A. Lyubimov, George Green: His Life and Works.
Usp. Fiz. Nauk, vol.164, 105 (1994); [Phys.-Usp. vol.37, 97 (1994)].

D. M. Cannell, George Green: Mathematician and Physicist 1793 - 1841. The Background to His Life and Works, 2nd edn. (SIAM Press, Philadelphia, 2001).