There are many books on specialized issues related to graphs: planar graphs, graphs on surfaces, graph coloring problems, distance in graphs, etc. However, if one is looking for a readable introduction that covers a lot of different aspects of "basic" graph theory (degree sequences, trees, colorings, matchings, connectivity, etc.), I think the best place to start is:
Introduction to Graph Theory, (Second Edition), Douglas West, Prentice-Hall, 2001.
