Given a simple graph G, the corresponding edge ideal IG is the ideal generated by the edges of G. In 2007, Ha and Van Tuyl demonstrated an inductive procedure to construct the minimal free resolution of certain classes of edge ideals. We will provide a simplified proof of this inductive method for the class of trees. Furthermore, we will provide a comprehensive description of the finely graded Betti numbers occurring in the minimal free resolution of the edge ideal of a tree. For specific subclasses of trees, we will generate more precise information including explicit formulas for the projective dimensions of the quotient rings of the edge ideals. In the second half of this thesis, we will consider the class of simple bipartite graphs known as Ferrers graphs. In particular, we will study a class of monomial ideals that arise as initial ideals of the defining ideals of the toric rings associated to Ferrers graphs. The toric rings were studied by Corso and Nagel in 2007, and by studying the initial ideals of the defining ideals of the toric rings we are able to show that in certain cases the toric rings of Ferrers graphs are level.