It is hard to recommend a book or paper without knowing your level of mathematical maturity, and what type of PDE you are interested in. If you are looking for a general introductory advanced undergraduate/graduate text, I would recommend
"Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems" by Randall J. LeVeque
It is a very practical book, but he does take the time to prove convergence with rates at least for some linear PDE.
The situation can become much more difficult for nonlinear PDE, and the results are scattered through research articles and monographs. A good general survey article covering a broad range of numerical methods (not just finite difference) is
"A review of numerical methods for nonlinear partial differential equations" by Eitan Tadmor
You can download the paper here: http://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01379-4/S0273-0979-2012-01379-4.pdf
For many (if not most) numerical schemes for nonlinear PDE, convergence proofs are simply not available. Sometimes this is because the PDE lacks a rigorous notion of solution and well-posedness theory (e.g., mean curvature motion of networks), and in other cases the right ideas for a proof have yet to be discovered (e.g., high order schemes for Hamilton-Jacobi equations, like ENO schemes). It's common for authors of papers to just check consistency of the scheme, and then check that it converges for some test cases.
If there is a particular PDE or class of PDE you are interested in, then perhaps I could refine my recommendation.