As many have already observed, the value of wonders depends heavily on the level at which you're playing and the strategy you are adopting.
As a large-map deity player fond of an expansionist and aggressive strategy as a democracy, I would have to name Hoover's Dam as the one I care about the most. By the time the Dam comes along, one should have a lot of cities already so the boost to production is substantial and very costly to duplicate otherwise (especially after allowing for the pollution). And the maintenance saving is also great. Even though one should be well ahead at this stage of the game, the boost is a great shot in the arm. If you still have a serious rival at this stage it is all the more important to build it first.
I think that Bach's Cathedral and Shakespeare's Theatre can be very important for the Democracy player for a reason that doesn't seem to have been mentioned here. Units away from home start causing unhappiness after the cumulative effect of improvements but before the effect of wonders. This means that, apart from the direct beneficial effect of making people happier, Bach's Cathedral can also be one of the few ways of keeping certain large cities both supporting units away from home and also in WLTP mode (so growing each turn). Wonders can do this in a way that city improvements simply can't. Shakespeare's Theatre is another way of addressing this problem: one can build the Theatre in a high-production (preferably coastal) city, then reassign the homes of all the bombers, AEGIS cruisers etc to that city where they don't make anyone unhappy. Therefore, I will usually aim to build either the Theatre in an appropriate city or Bach on an appropriate continent. I think it is worth building at least one of these oneself, to ensure they are in a suitable place.
Cure for Cancer, apart from the fact that it arrives rather later in the game, can also be built anywhere and is therefore equally well captured. So I make that a lower research/build priority.
If Bach and the Theatre are treated as one "either/or" rather than two, I guess I'd put Women's Suffrage third. Of course, one can build police stations in the appropriate cities instead -- but I find Women's Suffrage gives considerable tactical flexibility when expanding rapidly by conquest. Specifically, it makes it much easier to incite revolts and then actually use the newly acquired units (without the wonder you'll have to build the police station first or get the units to a new home base straight away which may be impossible). Also, it makes it easier to get newly conquered cities building further units faster, or taking over the home for existing units. And it saves a good amount of maintenance, if you'd expect to have a lot of police stations otherwise.
Why not the others -- when many people have quite correctly described their many virtues? Well, I must say that Adam Smith and Michaelangelo are also both extremely useful for my kind of game -- as many here have pointed out -- and I will also usually build these with some enthusiasm. But I tend to find that if the AI builds these earlier than I've gotten round to it for some reason, then it's not the end of the world. Assuming the AI CIV in question is not too large, then the benefit they get isn't too great and both can (and will!) be captured at the point where they become strategic.
The Statue of Liberty is another one I like and build when I can, but it's not crucial. I don't like to switch away from Democracy once I've got there, so avoiding the anarchy is nice but a bit of a luxury.
Magellan is also quite good -- those extra couple of movement points can make all the difference to getting in the first blow in a naval encounter, and make subs in particular a lot more dangerous. It can allow more cities to be connected via transports in a single move, which can be helpful. By using a carrier, it effectively can extend your arial range later in the game. And it also adds significantly to the power of a naval bombardment of a city. So this is nice, but I don't think it will turn the game for you, and can be captured if its important.
Turning to wonders that are less valuable, for me at least:
The UN is a bit of a two-edged sword. Personally, I prefer it when it's not built! I don't have trouble with my hawks and doves -- I find it fairly easy to provoke the AI to war if that's what I want anyway. Once at war, then I stay there until I'm ready to stop (often never) -- the last thing I want is the UN forcing ceasefires or peace treaties when I have the snivelling bastards by the throat! Once someone builds the UN, then you have the "imposed ceasefire" problem with that CIV, but only that CIV. But if you build it yourself, then you have the problem with everyone! (The same issues apply with the Great Wall -- but this also has the secondary problem that you have to build the city walls well before its effects expire anyway, otherwise you'll be caught with your pants down!)
Another couple of wonders that I don't usually build are Leonardo and Sun-Tzu, though I can see that there are lots of Leonardo fans here (and the movie is really nice). With Leonardo, the depressing thing is that it upgrades your veteran units to non-veteran replacements. Of course, you can look for appropriate battles to "toughen up" your green troops but this isn't always convenient. And then it expires just as things get interesting! I have a strong preference for veteran units, but Sun-Tzu still doesn't appeal to me much. I find that I still need plenty of barracks around for repairing units rapidly.
The Great Library can be of value, but often as a pre-emptive technique rather than a direct knowledge-builder. As others have pointed out, one would hope to be in the lead on technology. In deity mode that isn't always the case, and I have found myself sometimes building the Library either to keep up or, when ahead, to make it harder for other CIVs to keep up. But the other CIVs are pretty promiscuous in the knowledge trading anyway, especially when you're in the lead. And I usually just steal anything I need. And then the wonder expires anyway.
I don't have much else to add. The single city wonders are inherently limited (except Shakespeare's Theatre for the reason mentioned) and Pyramids is nice but may be over-rated as others have explained. The Manhattan project is just a pain in the neck but the AI will eventually build it if it gets the chance. The Oracle can be useful at the beginning, in a deity game, but it doesn't tend to last long. Same for the Lighthouse, though I can see that on certain maps that may be important anyway, even for a short time. I share the view that Darwin and Marco Polo are not worth the effort. And if you care about your reputation (as I do, for no very rational reason!) then the Eiffel Tower is probably better captured later in the game (to repair damage) then built earlier.
I hope these comments are stimulating!
All the best,
Steed
