For a general discussion of a simple yet powerful approach for implementing the finite-difference method, see Lecture 10 here:
http://emlab.utep.edu/ee5390cem.htm
For a quick discussion specific to Laplace's equation, see the notes for Topic 4 here:
http://emlab.utep.edu/ee4380microwave.htm
This second link derives Laplace's equation for electromagnetics, but the remainder of the lecture is just about Laplace's equation.
If you are interested in electromagnetic applications, you may want to take a look at a more advanced finite-difference approach described in "Transmision Line Analysis" here: