Wipe is short for wipeout, meaning a situation where the entire party or raid is killed. Sometimes referred to in other games as a "TPK" (Total Party Kill) or "TPW" (Total Party Wipe). Wipes can occur for many reasons — a bad pull, a group member failing to do their job, unexpected adds (e.g., from a patrol or a runner), etc. The term is rarely, if ever, used in PvP.
Wiping is actually a central element of the endgame PvE content. As described in Raiding for newbies, the boss mobs in the raid instances are very hard, and only very rarely will one of them be defeated on the very first pull. Normally, bosses have to be "learned" by the raid, which involves many wipes. Some bosses remain difficult even after they have been killed a few times, other bosses quickly gain "farm" status, which means they can be killed by that raid with little effort.
When a group wipes in an instance and they want to continue adventuring in the instance, the group has to walk back as ghosts to the beginning of the instance and resurrect, then traverse the instance to where they wiped. However, this is very time-consuming, and (especially in endgame instances) some mobs may have a short respawn time, thus the group may be forced to re-fight the way back to where they were before. Occasionally when something like this happens, groups will fall apart, especially in pickup group raids. Fortunately some classes have wipe recovery abilities which can significantly reduce the tedium associated with wiping, such as a shaman's Reincarnation ability, although more experienced groups will often prefer to use these during combat to increase their chances of success rather than mitigating the consequences of failure.
One of the most famous wipes was the Leeroy Jenkins wipe. (see video)
See the wipe recovery article for details.