Hierarchy or organizational charts represent the relationship between categories that implies a ranking or subset. For example, since Justin Time is head of Marketing and Will Knot is branching below him, that implies that Will Knot is also in marketing and works for Justin Time.
Some other examples that this is type of graph could be used for:
Department structure
Taxonomy (classifying organisms)
Evolution
Illustrating mathamatical concepts
and more!
Network Graph
Unlike the Hierarchal graph, the network graph does not necessarily imply ranking or subset. The nodes represent a value (usually a person) and the lines between them indicate some kind of relationship (e.g. co-wrote a research article, or facebook friends). The size of the nodes or connecting lines can add additional values (e.g. number of papers written/ written together). You can also add arrows to imply direction (e.g. follows on Instagram).