Literally, to behave as if someone, or some group, was your peer.
In telecommunications, peering is an arrangement whereby two parties (the peers) reciprocally exchange traffic (IP traffic or POTS traffic) with each other. Usually this traffic is exchanging without charging, on the assumption that each of the peers benefits from the relationship. Other less common forms of peering can include payment for traffic imbalances between the peers.
From an end-user point of view peering is beneficial as it often means that traffic reaches the intended destination earlier (reducing latency and jitter).
From an ISP point of view peering is beneficial as it reduces the amount of traffic passed on to your transmit provider (hence lowering the cost of that link).
From an transit providers point of view, peering is often seen as a problem as it reduces customers reliance on transit connections (hence lowering their revenue).
Traditionally peering happens between large players in the Internet market such as Telcos and ISPs, however there are also benefits for large sources or sinks of traffic to peer at a peering point for the same reasons as an ISP (it reduces their reliance on purchased transmit). Organisations that need to be multihomed may also choose to connect to their providers at a peering point.
NewZealand has two major peering points APE (Auckland) and WIX (Wellington) both of these are managed by CityLink.
An excellent introduction to peering is William B. Norton's paper "Internet Service Providers and Peering" downloadable from 
http://www.equinix.com/pdf/whitepapers/PeeringWP.2.pdf
Keep an eye out for other William B. Norton papers as well, they are usually highly informative and well researched.
Peering breaks down when peers are not honest brokers for their customers--widespread peering increases redundancy and reduces the pressure on long-distance links, but there are certain pricing schemes which do not provide incentive for peers to do this.
An apparent instance of this happening can be found here: http://www.nzherald.co.nz/storydisplay.cfm?storyID=3568316
http://scorchio.pure-guava.org.nz/cgi-bin/wiki/kwiki.cgi?PeeringNotice
http://scorchio.pure-guava.org.nz/cgi-bin/wiki/kwiki.cgi?DePeeringForNewbies
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