As part of the Master of Mathematics / Master of Advanced Study, candidates may choose to offer an essay. This will count for 3 units, or about a sixth of the course. There is no prescribed length for an essay, but the general opinion seems to be that 5,000 to 8,000 words is a natural length. The essay does not have to be original in content.
Each year members of the Faculty propose suitable topics; links to those for the most recent submissions are provided below (subsequent years are expected to be broadly similar, although not identical). Students are also free to propose their own topic (subject to confirmation by the Faculty Board of Mathematics). Note that if an essay is written on a particular topic in a given year then that exact topic can not be set in the next year.
A list of extra topics is generally offered during Lent Term, on the same basis and terms as the any on the initial list (listed as "Additional Essay Topics").
The primary requirement on the presentation of Part III essays is that they are legible. Hand-written essays are acceptable (if legible), but you may prefer to use the text formatting software which is available on the University PWF network.
I think Part III is quite common amongst US students nowadays, and is thus well-known to graduate admissions committees. Probably the effect on your application will be good, since if nothing else it shows you are serious about studying mathematics; a school can also reasonably expect you to be considerably better prepared coming after a year at Part III than when you finished your BA. So if one of your issues was a weak curriculum as a BA student (as opposed to poor grades or weak letters), then Part III could help quite a bit. On the other hand, I wouldn't count on a dramatic change in your graduate school admissions, in part because as you note, you won't have your grades or a strong recommendation from Part III in hand when you apply for graduate schools the next time around. I wouldn't worry about the fact that Part III is not research based; obviously doing research before starting graduate school is great, but most graduate schools in the US are not really expecting you to have done much in the way of it beforehand, or to be even close to ready to start when you enter.
I think if money is no object, then Part III is probably as good as anything else you might do for having a strong graduate application in the coming cycle. EDIT: I should probably also say that a matter of substance (as opposed to application strategy), Part III is probably on average better than spending a year as a grad student at a random respectable grad program in the US, since it will bring you into contact with a wide variety of other students and ideas.
However, I think you should weigh that next to the possibility of starting graduate school at one of your lower preferences, and then trying to transfer after a year, or when you get a master's degree. There's no guarantee this will work, but the same can be said of Part III. If you've been offered funded admission at a respectable school in the US, and would have to pay your own way at Part III, I would look hard at whether you think it's worth the money.
EDIT: I wouldn't count on the "glamour" of Cambridge itself to have a strong effect. I honestly don't know how selective Part III is (maybe someone who knows can comment.), so I wouldn't rely on assuming that admissions committees will consider it as such. There's some psychological "band-wagon" effect where getting one prestigious position reinforces getting others, but it won't work if the substance isn't there. Getting a BA from Harvard or MIT is helpful for getting into graduate school (if you have a strong record) because an admissions committee is more confident that getting an A in math class at Harvard really means something, and that a professor at Harvard has a lot of experience with talented undergraduates and thus can speak with some certainty about what it takes to succeed in graduate school. So, if you went to Part III, got good scores on your exams and got a strong recommendation from a professor there, that could strengthen your application a lot as a "second opinion" reinforcing the recommendations and grades you have from your BA. However, as discussed, those would only be available for the fall 2016 admission cycle, not 2015. One possibility is to see about accepting one of your safety schools with a deferment to go to Cambridge, going to Part II in 2014-5, going to the safety school in 2015 and applying for transfer in 2016 or 2017 if you're unhappy there.
One piece of information we're lacking is what your "lower preferences" are (don't put too much stock in USNW rankings, but they help to be concrete in a discussion like this). It makes a big difference whether we're talking about a school in the rank ~25 (like UCSD), ~50 (like UVA) or ~100 (for example, South Carolina). In the former case, I'd say it's a waste of time to try to try to move up unless you go and are miserable, whereas in the latter it makes a certain kind of sense.
LATER EDIT: Incidentally, yes, given that your undergraduate grades are already "baked in," the main things you can still hope to change are your letters, and also your GRE scores if those were bad.