Networks are one of the most common ways to represent biological systems as complex sets of binary interactions or relations between different bioentities. In this article, we discuss the basic graph theory concepts and the various graph types, as well as the available data structures for storing and reading graphs. In addition, we describe several network properties and we highlight some of the widely used network topological features. We briefly mention the network patterns, motifs and models, and we further comment on the types of biological and biomedical networks along with their corresponding computer- and human-readable file formats. Finally, we discuss a variety of algorithms and metrics for network analyses regarding graph drawing, clustering, visualization, link prediction, perturbation, and network alignment as well as the current state-of-the-art tools. We expect this review to reach a very broad spectrum of readers varying from experts to beginners while encouraging them to enhance the field further.