The basic difference between a traditional telecoms network and a mesh network is that the traditional kind consists of large, expensive central beacons or masts that each user device (like a mobile cellphone) connects to, while a mesh consists of scattered (small, cheap) nodes, each of which can connect with a number of others. A message spreads from one part of a mesh to any other part by passing through as many nodes as necessary.
In a traditional network, the beacons are usually connected to each other (and the world) by a “backbone” landline or microwave radio link. A mesh network may or may not have a “gateway” which is just one of its nodes which is also connected to a traditional network. Thus if one member of a mesh has an internet connection, it can be shared by the whole mesh. But the nodes still connect with each other when there’s no gateway – e.g. the Mankosi community can phone each other for free even if their internet connection is broken.