Network analysis is the study of relationships. A bunch of things, at least some of which are connected, makes a graph; when the connections vary in strength, you have a network.
In ecology there are two main uses of networks: studying spatial relationships, or social ones. This is a very brief run-through of how I got started looking at spatial networks of bird movements in R.
(If you’re more interested in social networks, check out this paper from Ferreira et al. on getting started, and this one by Silk et al. for the CMRnet R package to implement it)