What. Network Graph are composed of nodes (individuals) and edges (connections). Nodes that are highly connected appear closer on chart. Networks can be directed or undirected, showing a connection or a flow.
Why. Network diagrams allow to visualize relationships between a group of entities, describing how things are interconnected. Customizing nodes appearance allows to compare with an expected structure.
How. Several R libraries are available. Igraph is the most important allowing to make every type of customisation. NetworkD3 allows to draw interactive network, ggnet2 and ggnetwork are other interesting libraries.
About Network Graphs
The Igraph Library
Interactive Network with network D3
The GGNET2 library