What is the difference between galerkin and discontinuous galerkin methods.

Classic Galerkin enforce continuity from one element to the next one. In each node, a variable has a unique value. If you use linear elements, there is continuity of the variable solved for, but not of its derivative. Using higher order elements, you can enforce that.

Discontinuous Galerkin doesn't enforce such continuity between elements. The method allows for different values in neighbouring elements at the same physical location. Usually though, for good accuracy, the jump across an interface is (very) small. In the end there has to be a way to couple the solution in the different elements, this is done by solving a Riemann problem.

What paper you may offer for the first acquaintance?

A good introduction would be the online course of Tim Warburton. http://www.caam.rice.edu/~timwar/MA578S03/MA578.html

If you want a single file to look at you read the notes from Bernardo Cockburn. http://www.math.umn.edu/~cockburn/LectureNotes.html

Both are DG experts.