While I appreciate Brian Ballsun-Stanton's explanation in so far as it cites the concepts of Game Theory that arose in the 1930s and '40s to explain rational decision-making in economic systems, I believe that it fails to capture the emotional thrust imbued in the notion of "Overpowered." Games Theory is primarily concerned with dominating strategies - that is, the better choice to make.
The problem with the notion of something that is Overpowered is not that it is the better choice, but the magnitude of difference between that choice and other choices. Moreover, if it were merely a better choice, something being Overpowered would be a good thing and everyone would take it. However, most people use the term Overpowered in a negative sense: suggesting that the Overpowered thing should be removed from the game.
Games Theory would suggest that most people would want to be Overpowered and would therefore want Overpowered as an option within the game. This does not correlate with the reality that most people want Overpowered options removed from games.
Therefore, I believe a much earlier principle better explains the notion of Overpowered, the theory of Justice espoused by Ancient Greek Sophists, though perhaps best worded by John Rawls in his essay Justice as Fairness.
Justice is thought of as a pact between rational egoists, the stability of which is dependent on a balance of power and a similarity of circumstances.
Players of a game expect a similar sense of justice to prevail within the game environment. Indeed, saying something is Overpowered is a subset of saying it is unfair. Fairness, therefore, seems to be the primary principle leading to the negative connotations behind Overpowered.
In order to be fun, a game must have a certain degree of stability, balance of power, and similarity of circumstance, just as John Rawls identifies as cornerstones of Justice. The basic principle in most Role-Playing Games is the expectation that all players will be able to create characters by following the standard rules of roughly equal power.
"Roughly equal power" is a difficult concept to define. If one character rolls 1d8 damage and another rolls 1d10 damage, is that roughly equal? What if it's 1d6 and 1d12? 1d4 and 2d6? All of these are damage dice that show up in weapons in D&D 4th edition, and all of these are considered roughly equal, at least in so far as the character dealing 2d6 isn't considered Overpowered by the character dealing 1d4.
On the other hand, a character rolling 100d12 damage would be labeled as Overpowered. There is therefore some unknown tolerance factor that we have to assume exists but cannot strictly define; this also means that we cannot strictly define whether an element is Overpowered. Instead, we merely know that, for any element that violates the tolerance threshold, that element is Overpowered.
The tolerance factor will vary both by game system and by player, further obfuscating the nature of Overpowered. Something might feel Overpowered to one person but not feel Overpowered to another, or something that is generally considered Overpowered in one game system might not be Overpowered when similarly presented in another system.
Rawls has more to add on the notion of fairness that might help us better understand Overpowered:
Usually acting unfairly is not so much the breaking of a particular rule, even if the infraction is difficult to detect (cheating), but taking advantage of loop-holes or ambiguities in rules, availing oneself of unexpected or special circumstances which make it impossible to enforce them, insisting that the rules be enforced to one's advantage when they should be suspended, and more generally, acting contrary to the intention of a practice.
Emphasis added by me. If this hadn't been written in 1958, I'd be certain Rawls were referring specifically to D&D :)
The key issue here is the notion of the intention of rules of a game. This leads to the distinction of RAW (Rules As Written) and RAI (Rules As Intended). Often, something that is Overpowered follows RAW, but violates RAI; or so players complaining about the Overpowered thing might claim.
For example, in my lengthy discussion on A Brief History of Basic Attacks, I argue that overriding the Melee Basic Attack with at-will powers in D&D 4th Edition violates the intention of a Basic Attack being a Basic source of damage without the same level of crippling effects present in At-Will powers. A Fighter using Beast Switch as an MBA, interrupting an enemy's attack by sliding them away and knocking them prone and thereby cancelling out that attack, is considerably more powerful than simply dealing damage, and in violation of the intention of Combat Challenge's damage reprisal for ignoring the Fighter's mark.
However, this is merely one interpretation of the intention. Did the game designers intend for the Fighter to merely deal damage (as I claim, by comparing that mark reprisal to other damage-only effects like the Paladin's Divine Challenge or the Battlemind's Mind Spike), or did they intend for it to nullify the attack (as others claim, either by suggesting that building complexity is a feature of the game or by comparing it to Swaying Branches, which allows Wardens to forgo dealing damage to potentially nullify an enemy's attack by sliding it).
Both sets of arguments depend on how a player interprets the intention of the rules, and really this is at the core of the notion of Overpowered as well. The reason there is no way to come up with a self-contained definition of Overpowered that can be used to identify Overpowered things in any setting is that Overpowered is a subjective, rather than objective, property. It is as difficult to strictly understand as beauty: most people simply know it when they see it.
In conclusion, I would suggest the following answer:
Something is Overpowered if it violates the intention of the game system.
However, this is merely an objective short-hand for the subjective reality behind the notion, so it would be more completely defined as:
Something is Overpowered to you if you feel it violates the intention of the game system.
How does this account for games like World of Synnibarr and RIFTS, that clearly contain overpowered things, but these things are congruent with the intentions of the system? Perhaps the it has something to do with balance not being part of the design intentions? If so, does this answer only apply if we first assume balance as a necessary good?
If the powers in RIFTS and other similar systems are intended by the game's designers to have a large variance in power, these powers are, by my definition, not overpowered. They are not OVER powerful because they are not powerful OVER the power intended for them to have. They may be "very" powerful or "too" powerful but, whatever they are, they aren't "overpowered."
The term "overpowered" carries a negative connotation: if something is overpowered it is, in some sense, bad. In my analysis, this badness comes from a violation of intended game balance. In order for violating the game balance to be bad, respecting the game balance must therefore be "good." Thus, in order for someone to be able to categorize a feature of a game as overpowered, they must also be assuming that balance in that game's system is good.
Thus, while I am not suggesting that every game system must be balanced or that every player must view balance as necessarily good, a player must view balance as a good in order to be able to discuss overpowered aspects of a system.