A peer-to-peer (P2P) network is a group of 2 or more devices – for example, mobile devices, computers or even printers – that are set up in such a way that they can share certain data with each other. There's no central server acting as a shared drive where data is stored. Instead, each computer acts as the server for data to be stored.
A "peer" is a computer system on the network. Each peer is considered equal and they are commonly referred to as nodes.
Say Alice wants to share a file with Bob. In a non-peer-to-peer network, Alice would typically give it to Bob through a central server, to hold. Bob would then have to request access from the server, so he could use it. In a peer-to-peer network, Bob doesn't have to do this, it goes straight from Alice to him.
Uber is an example of a non-peer-to-peer network. When you book an Uber, you're contacting Uber for a driver. When you flag a taxi down, you're interacting directly with the taxi driver. Uber acts as the central server, if their network goes down - you cannot book a driver. On the other hand, a peer-to-peer network means that if the server goes down, it can't take the whole network with it.