Basic Definition and Benefits
Mesh networks are networks that are, in a sense, "meshed" together. In a typical network, many nodes are connected to one gateway, typically a router, that connects those nodes to the Internet. These nodes are usually not connected to each other in any real way. To contrast, a mesh network maintains connections between all the nodes in said network. Because many networking implementations make extensive use of graph theory, it is prudent to sometimes inject graph theoretical terminology to describe certain network topologies. A mesh network is a connected graph, whereas the aforementioned "typical" network is more likely a star graph (bipartite).
The benefit of this node connectedness is that if one connection fails between nodes, those nodes can still connect by taking another route (there are cycles in the graph). This allows for high network dependency. This also dovetails into another important positive aspect of mesh networks: they are decentralized. If one node in the mesh fails, the network stays intact.
Other benefits include ease of and relatively low cost of implementation, particularly suiting this networking strategy to rural areas that are deemed too risky/expensive for ISPs to provide network infrastructure for.