What is the difference between a peer to peer network and LAN?

A Local Area Network (LAN) is typically a private network that is only visible behind your router or firewall. It is typically an TCP/IP based network with a NAT-type Firewall and/or Router that shares an IP address that is visible on the internet (sometimes referred to as a Wide Area Network or WAN).

An example would be two computers behind a residential router. The network between devices that allow them to communicate would be the Local Area Network.

A Peer to Peer (commonly abbreviated as P2P) network describes a type of ad-hoc network that typically allows the ability to search for and download files without having a specific central server.

A Peer to Peer Network and a Local Area Network are actually quite different. To put it in an analogy, a Local Area Network is like the postal service. It lets you send messages to other people. The Peer 2 Peer network would be what you send in the messages and how you find other people to share with.