In this dissertation we study the homological algebra of the monomial ideals with a special emphasis on the topics of the Castenuovo-Mumford regularity and the powers of edge ideals of finite simple graphs. The main problem of this dissertation is to find optimal bounds for the regularity of powers of edge ideals. To do this, we prove the existence of a very special order of the minimal monomial generators of powers of the edge ideal. Using this order and some short exact sequence techniques we prove that the regularity of a power of an edge ideal can be bounded by the maximum of the regularities of the edge ideals of some very closely related graphs, and as corollaries we show that for various classes of graphs the higher powers of edge ideals have linear minimal free resolutions. One of these corollaries partially answers a case of a conjecture proposed by Eran Nevo and Irena Peeva. In the process of this study we introduce a new notion called even connectedness in finite simple graphs and derive various results related to it. In particular, we show that this behaves particularly nicely in the case of bipartite graphs and prove some results related to regularity of powers of edge ideals of bipartite graphs. We also study path ideals of finite simple graphs in the same spirit and show that various classes of path ideals also have linear minimal free resolution. Using similar techniques we also study the Cohen-Macaulayness of bipartite edge ideals and prove a new characterization for it.