How does funding at UK universities work? The process for obtaining funding for graduate work in the UK seems much different than it is in the US. In particular, in the UK, funding decisions often come after offers of admission. What is the process by which a good, accepted graduate application at, say, Oxford, is given funding?
The funding depends on your nationality/residence by a lot, as well as the field, the university, and the supervisor.

UK or EEA residents can get funding from national research councils (such as ESPRC, STFC and many more) which give them (usually) the cost of tuition plus a small living stipend (when I was applying the living wage was about £13,000).

Non UK/EEA people generally have to self-fund outside of a few special circumstances (competitive international scholarships such as Rhodes, university bursaries for internationals). This is very very rare. However, it's not terribly uncommon to see students from an above-average financial situation self-funding, but then they need to pay not only their living costs, but also the cost of tuition (at my old university it was about £20k per year for international students).

I wrote an extensive response on all of these issues here and here and here. The other answers from other users on those posts are also excellent.

So what happens is that you apply, and then you will be (hopefully) asked for an interview. If you are able to compete for the 'normal' funding pot, you will be interviewed much the same way as everyone else. First a shortlist, then the interviews, and maybe follow-up interviews will be used to select the best candidates from the pool of applicants. If you're non-UK/EEA, you may or may not interview with everyone else; it's up to the university. If they don't have the money to fund you, many departments will make you an offer anyway (especially for foreign students) which you will usually have several months to accept or decline. 

The research councils will grant the department a certain number of PhD studentship bursaries, and they will distribute them. From what I've seen, you don't have to do much self-application for scholarships unless you're foreign. Though you will always be told where the money is coming from, and there may be some requirements (a yearly seminar, for example) to get the money.

If you are given a stipend, it is tax-free, unlike in many other countries. Your funding is all-inclusive, not split up into RA or TA like in the US. You can normally make extra money marking coursework or demonstrating in labs, although the 'estimated number of hours' taken by these activities doesn't always match up with the actual number of hours you spend on them !