I am trying to answer from an electromagnetics perspective, although the basics are same for other applications.

a) FEM discretizes both the source as well as solution regions under consideration. Typically discretized using tetrahedral volume mesh elements in electromagnetics. Maxwell's equation in its second order differential form (Helmholtz equation) is solved for. Directly solves for electric or magnetic fields.

b) BEM discretizes the surface or boundary of the source. Typically discretized using triangular surface-based mesh elements in electromagnetics. It is an Integral equation based method which captures the physics in the form of Green's functions. Solves for charges and currents on the geometry, from which the fields can be post-processed.

c) FDM works well on regular grids and "not-so-irregular" geometries. The first order derivatives are approximated as finite differences from Taylor series approximations. Second order equations in the form of Laplace and Poisson's are also solved in electromagnetics, subjected to known boundary conditions.

Very shortly, FEM presumes space discretization of bodies under study, while BEM is based on their surface (contour) discretization only. For example, in structural mechanical analysis, the different kind of meshing by these methods produces different type of matrix in relations of stress and deformational factors. According to BEM we may use analytical functions which relates boundary stresses and displacements.  Finite difference method presumes discrete form of writing of differential equations which described stress-strained state of body.

The important peculiarities of BEM in comparison with other discrete methods are shown good in classical books and papers of  Brebbia C., Krauch S. and Starfild A.

All the best

Dear Shrish,

Here are some links that may answer your question

https://en.wikipedia.org/wiki/Boundary_element_method

http://files.campus.edublogs.org/blog.nus.edu.sg/dist/4/1978/files/2012/01/CN4118R_Final_Report_U080118W_OliverYeo-1r6dfjw.pdf

http://wwwf.imperial.ac.uk/ssherw/spectralhp/papers/HandBook.pdf

 With my best regards

Prof. Bachir ACHOUR

 FDM finds the solution at points, and works well on regular geometries. FEM method, atleast from the galerkin perspective, uses basis functions to represent the solution to the differential equation. The finite element method finds the coefficients on these basis functions to represent the solution.  FDM has stronger requirements on existence of derivatives of solutions, and the stability of numerical solution, but is difficult to implement on irregular geometries. FEM, while not as simple, works well on irregular geo. and gives a series solution to the differential equation. 

I am trying to answer from an electromagnetics perspective, although the basics are same for other applications.

a) FEM discretizes both the source as well as solution regions under consideration. Typically discretized using tetrahedral volume mesh elements in electromagnetics. Maxwell's equation in its second order differential form (Helmholtz equation) is solved for. Directly solves for electric or magnetic fields.

b) BEM discretizes the surface or boundary of the source. Typically discretized using triangular surface-based mesh elements in electromagnetics. It is an Integral equation based method which captures the physics in the form of Green's functions. Solves for charges and currents on the geometry, from which the fields can be post-processed.

c) FDM works well on regular grids and "not-so-irregular" geometries. The first order derivatives are approximated as finite differences from Taylor series approximations. Second order equations in the form of Laplace and Poisson's are also solved in electromagnetics, subjected to known boundary conditions.