You can solve amazingly large problems with a sparse matrix with out understanding what you are doing, as long as you use direct (factorization) methods.  If you type A\b with some gigantic sparse A, matlab will apply reordering heuristics that often just work, and are actually incredibly hard to improve upon by manual fiddling.  When you get to really, <i>really</i> big problems, you need to use iterative methods (like conjugate gradient), and for those you have to have a <i>very</i> firm understanding of everything.  The gap between direct and indirect methods is huge.