A peer is a another computer on the network.
The standard network system pushed by Microsoft is 'client/server' where the users computers are 'clients' requesting data from a central computer: the 'server'. Alternatively you can have a 'peer-to-peer' as used by Gnutella where all the computers are of equal status and are both clients and servers. So, when you are logged on to Gnutella, your computer is one peer, downloading from, or uploading to, other peers.