This is the easy one: Power is simply the ability to make another person (or group of people) do as you wish them to, either through force, threat or other overt forms of coercion or through more subtle means, such as by convincing them that they wish to do so.
Examples of Power being exercised publicly in the world today are so easy to find as to make it almost a pointless exercise: the U.S. campaign in Afghanistan, Microsoft's campaign to eliminate MP3 and other "uncontrollable" formats by blocking support in their operating systems, the entertainment industry's ability to force a retroactive extension of copyright and the passing of the DMCA on a table greased with industry money, Pacific Gas & Electricity's ability to hold almost the entire state of California hostage to their profit margin. I could probably go on listing examples infinitely, as new ones crop up faster than I can type.
The more subtle exertions of Power are much more difficult to list, naturally, as by definition they are much more difficult to detect and define. A recent well known example (to a web-based readership), however, would be Microsoft's failed attempt to force the DoJ to drop their suit by simulating a ground swell of popular support for Microsoft (these sorts of things are much easier to identify when they fail). Microsoft, by spending huge amounts of money, attempted to make it seem as though there were a popular upsurge of outrage against the DoJ case by arranging to have a Public Relations firm send carefully designed letters to newspapers and politicians and found several political groups which seemed to be an outgrowth of popular demand until it was realized that their addresses were the offices of Microsoft's P.R. company. This tactic is so widely practiced now that it's even got its own name, "astroturf", and should be quite familiar to any of you who have followed the legal shenanigans around the tobacco industry.
Just as there are whole industries dedicated to servicing the needs of overt power (police and military forces, "defense" contractors, "security" equipment and staffing), there are whole industries which service the needs of covert power (advertising, lobbying, "public relations").
The second term is much more difficult to define. This is natural, of course, since it's the one whose meaning has largely been effaced by its conflation with Power. Modern dictionaries are little help, except in that they list the historical root of the word: the Latin "auctoro", defined by the Oxford Latin Dictionary as to bind or obligate, a telling linkage to its rather older relationship to contracts. This is also unsurprising, as dictionaries track common usage and it's common usage which has started to make these words synonymous.
A more fruitful source, though, is another word which shares this root but has retained its independent meaning; in this case I am thinking of "authorize". "Authority" once meant "having been authorized", though now (to quote the primary definition given by dictionary.com) it is said to mean "The power to enact laws, exact obedience, command, determine, or judge". That's a rather significant drift in usage, though not in practical terms as historically it was a person in Power who would authorize another. The change is only really meaningful in the context of a State which claims to derive its Authority from the will of the people residing therein.
It's certainly easy to see why someone with Power derived from other sources than popular authorization (such as wealth, police or military affiliation, or control of some important resource) would want to conflate these two terms: it grants an air of legitimacy to their position to call their actions exercises in Authority rather than Power.
Power legitimizes itself; legitimate Power is simply Power sufficient to achieve the aims of its controller. There's not a lot to talk about here.
The issue of legitimacy for Authority, however, is much more complex and hence much more interesting.
In modern American usage, Authority derives legitimacy only from tradition (age) or from another, greater, Authority. The state and federal governments are commonly considered to hold legitimate Authority because they have been so considered for over 200 years and they are commonly considered to be able to delegate legitimate Authority to people and other organizations.
This is clearly inconsistent with both itself and reason. To allow an Authority to delegate its tasks by creating other Authorities may be reasonable (though, of course, that's a decision to be made by those who grant it its Authority), but to say that Authority may derive its legitimacy from tradition is certainly not.
First, it is either inconsistent or leads to an intractable problem. If Authority may be legitimized merely by the age of an institution, then naturally the American colonists were not entitled to create a new State, as this would be illegitimate rebellion against a legitimate Authority (the British State, which had at that time existed for several hundred years). Thus, the modern American State would be illegitimate and its claim to legitimacy through its age would be void. Additionally, if one were to argue that, though the American State was illegitimate at its inception, its current age serves to legitimize it a new problem arises: at exactly what age can you then say the state became legitimate? This is the same problem faced in the decision at what age a fetus or infant should be considered human. This question has not been demonstrated to have a definitive answer, and we have no reason to suspect that the analogous question of a State's legitimacy would not prove to be equally intractable.
More importantly, though, by accepting age as a legitimizing factor for Authority you deny anyone born or living within its claimed boundaries any right to participate in its creation or control. A State which exists and acts without the consent and endorsement of the people exists and acts solely on Power, not Authority. How could it, when the people do not consent to its existence or endorse its actions? To say that an Authority, by virtue of its age (which is, essentially, to say by its existence), is legitimate is also to say that any Authority which exists is legitimate. This, once again, conflates Authority with Power, for nothing more than Power is needed to establish an institution and see to its continuance.
The act of consent, or granting authorization, to legitimize an Authority must have three attributes. It must be a) active, b) free and c) informed; without all three of these, it cannot reasonably be considered legitimizing. To take these in sequence:
a) active consent
There is an example here which is probably familiar to most of you. There's a Monty Python sketch in which a group of British soldiers must make a run for it to get back behind their own lines, but they only have enough supplies to take all of them but one. The officer suggests, because one of them has no arms, that they decide who shall escape by a show of hands. There's a similar example in Red Dwarf where, in trying to convince Kryten that three skeletons are actually dead, one of them suggests that everyone who's alive should raise their hands. To allow consent to be passive (i.e. by claiming that merely by not leaving the area where a person was born and lives that person is giving consent to ruled by whomever claims Authority over that area) is essentially the same as these examples; it is to grant automatic legitimacy to any claim to Authority. This is clearly unreasonable, and is certainly not in line with common practice (i.e. you would consider it unreasonable were I to claim that you were obligated to a contract merely by stating that it exists and that you must move out of your house or be bound by it).
b) free consent
Coerced consent, or an act of consent made under threat or duress, also cannot reasonably considered legitimizing. This would amount to stating that any claim to Authority is legitimate, provided that it were backed up with sufficient Power. To do so would, again, not be in line with common practice either (i.e. it would not be reasonable to consider your giving me your wallet, because I threatened to shoot you if you didn't, to be a binding transaction).
c) informed consent
This one should be the most obvious. Consent given to one thing cannot, through misdirection or misinformation, be reasonably considered legitimizing. This is also in line with common practice (i.e. were I to convince you to sign a document by misrepresenting its contents, then it would not be reasonable for you to be bound by it).
These three are really more in the way of expounding on the definition of consent than in being qualifiers of it, since all three merely illustrate how we use the term and all three restrictions are completely consistent with the common practice of consent.
Here are a few conclusions which can be drawn from this consideration of these terms:
1) The American State is not a legitimate Authority, but rather merely enforces the will of its rulers by its Power. This is true because it has never, even at its creation, gained the consent and endorsement of the people residing within the borders it claims. Though it may be possible that it would receive a majority vote in its favor (though whether a majority would even turn out for such a vote is doubtful), this has never even been attempted. At its inception, it did not even attempt to poll the vast majority of those living within its borders.
2) Rusty's Authority over Kuro5hin is legitimate, on the other hand, as at its inception he had the express support of all of its users, and all new users have joined with full knowledge of how the system is operated (i.e. the editorial policies are public and in an easily accessible location). In fact, when he tried to step down, a significant number of the comments demanded that he stay in his current position.
I think it's fair to say of these conclusions that 1) was what I had in mind as I sat down to write this, and that 2) is a rather silly example, but it is illustrative of a major difference between the Net and the Real World. On the Net, we are capable of creating wholly new spaces and thus are not often faced with questions of legitimacy for Authority over those spaces. As the creator, Rusty is free to design a social space which is structured as he sees fit and as others are free to participate or go elsewhere (with no significant impact upon themselves), then his Authority within his space is entirely legitimate. In the Real World, of course, all of the space has long since been claimed so the outlet of creating anew is blocked.