Simple network, described above.
Acyclic networks have no cycles. A Directed Acyclic Graph (DAG) is a well known sub-type.
Hypergraphs are sets of elements with relations that include more than a pair of elements (i.e. they are members of a higher cartesian product).
Hypergraphs can equivalently be represented as Bipartite Networks, where there are two types of edges (a special case of a multipartite network, where there are many types). On the other hand, a multiplex network is one that has multiple types of edges.
Trees are connected (can reach all vertices following edges), undirected networks that contain no closed loops. A forest is a disconnected graph whose connected parts are trees.
A Planar network is a network that can be drawn on a plane without having any edges cross. It is a special case of a Spatial network.
Temporal networks are those for which the set of edges and/or nodes varies with a time parameter.
A Similarity network is one that expressed how similar entities (expressed as the nodes) are. The degree of similarity being the weight of the node.