An edge cover of a simple graph is a subset of the edges so that each vertex is incident with at least one edge in the subset. The edge cover polynomial of a graph is the generating polynomial of the number of edge covers of the graph with k edges. The number of edge covers of path graphs form the Fibonacci sequence, while those of cycle graphs form the Lucas numbers. In this paper, we first provide some known and new general results on edge covers and edge cover polynomials. We then apply these results to find the number of edge covers and the edge cover polynomials of caterpillar graphs, cycles with pendants, and spider graphs, which are generalizations of star graphs. The number of edge covers of all these families can be expressed in terms of Fibonacci numbers. We generate new sequences not in the On-Line Encyclopedia of Integer Sequences using these edge covers and provide new combinatorial interpretations of some known sequences.