What is the difference between Finite Difference Methods, Finite Element Methods and Finite Volume Methods for solving PDEs? Can you help me explain the basic difference between FDM, FEM and FVM? 
What is the best method and why? 
Advantage and disadvantage of them?
This is a difficult question to answer.


  "The FDM is the oldest and is based upon the application of a local
  Taylor expansion to approximate the differential equations. The FDM
  uses a topologically square network of lines to construct the
  discretization of the PDE. This is a potential bottleneck of the
  method when handling complex geometries in multiple dimensions. This
  issue motivated the use of an integral form of the PDEs and
  subsequently the development of the finite element and finite volume
  techniques."
  (http://www2.imperial.ac.uk/ssherw/spectralhp/papers/HandBook.pdf)


Here are two references to review so you can get a better feel for these methods.


http://files.campus.edublogs.org/blog.nus.edu.sg/dist/4/1978/files/2012/01/CN4118R_Final_Report_U080118W_OliverYeo-1r6dfjw.pdf (see page 10 for a very nice comparison in the types of problems they were interested in - computational fluid dynamics)
There are some nice references for these methods at http://www2.imperial.ac.uk/ssherw/spectralhp/papers/HandBook.pdf (See section 7 for very nice references)
finite difference method is the oldest method to find the limited region or close region and FEM is the structural method to solve the partial deferential equation.