A user which understands the software and/or computers quite well. 

Although they are easy to train, in some cases they can grow scary when they start to "tweak" stuff in ways not anticipated, risking the integrity or predictability of system behavior. At that point they morph into a "franken-user".

''Generally (though not always) differentiated from "programmers" on the same systems.''

I differentiate programmers from power users by knowledge domain. Programmers know what it takes to make software, while power users know the range of software out there. It's the difference between knowing how to make wine and knowing the wines that exist. One can know some of both, but nobody knows all of both.
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* From my POV the PowerUser is a person who uses a system intensively. They know a multitude of usages and programs or applications and use them 'all'. They do not fall back to non-computer means so easily in order to accomplish any given task. For example they use a graphics tablet to input their signature, have their DVD collection organized by a database, use a voice over IP telephone, FAX if necessary directly from word processor via ISDN-card, attach the HiFi system to the PC output and have the whole music collection on disk, synchronise the filofax on their smart cellphone with the local address book and calender, play network-games with friends for recreation and real networking, host their own homepage on their own local server and what have you. 
* I do not see any reason to state, that a PowerUser would not engage in programming. They at least do scripting to turn the PC into an alarm clock to wake up with a variety of favourite songs. I's rather suppose that PowerUser connotes a more private use of any given system. a professional use of a privately owned system which easily consists of a good number of connected devices of all kinds in the sense of HomeAutomation. So the PowerUser is like competent and engaged SuperUser of a household. --ManorainjanHolzapfel

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See also: PredictionAndControl, CompilingVersusMetaDataAid, BusinessRulesMetabase