Network analysis is a method of information visualization and analysis that lets a scholar look at the connections between certain entities. An example might be a graph of interactions between characters in a movie. Through this graph it could be seen how many interactions are between a given set of characters, which interact the most, which the least, and what characters may be a bridge between different networks of characters. You can also do this for things like financial or organizational connections.
A network analysis has two basic parts. The first are nodes, also called actors or vertexes depending on the language of the software you're using. These are the people, places, organizations or concepts that you are interested in. The second are links (vertices or edges in some programs) and these are the connections between those different nodes. The nodes can be different colors or shapes depending on different categories they fit in. The links as well can be different colors or shapes.