A term which originates from MUDs which describes unproportional fast level advancement of a player character, quicker than the game was designed for, or even expected by the designer. This is usually done by experienced players who want to raise a new character quickly.
In general, power leveling usually involves collecting appropriate items (usually the best items without level restrictions, so the newbie use them) and ample cash/gold for the new character. Naturally this character will have an edge over "normal" newbies, and be able to collect experience very quickly and thus level up much faster than usual.
To speed up the advancement even more, often the newbie is supported by one or more high-level characters (sometimes by a whole party), which support him with spells, healings, transportation (portals), etc. Often big mobiles (aka mobs) are almost killed or disabled by the supporting party, then the newbie walks up and finishes them off to collect some big xp
Power leveling is not cheating, at least, not technically. DikuMUD, for example, features some more or less effective xp splitting rules which will naturally give more of the earned xp to the character who has the higher level. However, this can be circumvented very easily by simply not grouping.
How well power leveling works heavily depends on the MUDs policies on grouping, and even more on level restrictions on "strong" items.
If power leveling works well, literal races who would be able to beat the record to max out the levels of a character can take place.
For some comparison, on the MUD I used to play on (which happened to be powerleveling-friendly), while it took me about 6 months to max out a character without power leveling and any previous mudding knowledge, I've seen the same type of character reach the highest level within 2 days if power leveled.
All this stuff of course not only applies to MUDs only, but to any other similar level-based multiplayer games, like Diablo, Everquest, Ultima Online, etc.
As for Everquest, I personally know somebody who does "power leveling for fun and profit" by leveling up characters quickly and then selling them off on eBay.