Generally speaking, the terms 'neighbor' and 'peer' are interchangeable. The word 'peer' originates from the concept of peer-to-peer networking. A neighbor or peer is a router with which you have established a BGP session with. Neighbors and peers exchange BGP updates according to their administrator-established routing policies.
Peering is when you establish a BGP session to another BGP speaking device.
* SPECIAL NOTE *
Be careful with the use of the word 'peer' or 'peering' when talking to the backbone providers like Sprint, UUNet, Cable and Wireless, Qwest etc. To them, the word PEER has an entirely different meaning.
To them, a PEER is another backbone provider, everything else is a customer session, and therefore is NOT a 'peer'. I've had blockheads at UUNet say "We don't have peers, we ARE the Internet" and then hang up on me when I describe my BGP session as a 'peering' session.