For Oxford (Cambridge may be slightly different - check their web site):
1) Meet basic quals, which they state on the web site are 700+ M&V on SATs, top grades, and lots of SAT IIs and APs. They stated that (for 2004) they got 270 US applications and accepted 20% of those students. [They DO care about ECs, but not in the same way as the top US schools. It's not necessary, to be, for example, the absolute top baton twirler or center forward in your state, just that you have 2-3 interests that are well developed so that you will be likely to enjoy yourself outside of class and add to the life of the university].
2) Get a fabulous reference from your school head or guidance counselor. There is only one required.
3) Write a succinct personal statement that is clear about why you want to study the subject(s) and relates your current or recent past studies to the course (degree).
4) Remember the early dates! Pick Oxford or Cam and get everything to them by no later than October 15.
5) Send all SAT, SAT II, AP scores, and college transcripts (if any) directly to the admissions office.
6) Depending on the subject, you may need to send samples of graded written work - select work that demonstrates your analytical abilities. Print out the cover sheet from the admission web site and allow plenty of time to get the teacher to certify the paper and to mail it by the deadline.
7) If you are in North America, there are interviews offered in November in NYC or Vancouver. If you are invited for an interview, you have made the first cut, and it is mandatory. If in another country, you will either have to go to Oxford or check the web site for information (I don't know the arrangements).
8) You will sit a test in your subject on the same day as every other applicant. You have to arrange for a school official or an official testing center to proctor you and they must fax the test to the admissions office.
9) You will also need to apply through UCAS, the central online service for UK university admissions - here is the link for internationals. http://www.ucas.com/studyuk/index.html
You can apply to up to five additional universities, so look at the London Times lists for your subjects and select several that suit your tastes and are well-reputed in your subjects.
10) You will receive notification by the last week in December. If you get an offer, you will have to meet the conditions specified on the offer letter.
11) You can then select Oxford as your first choice in UCAS and also select an "insurance" choice (in case you don't meet the offer conditions but do meet the conditions of another university, for example 555 in APs versus 554).
12) When you meet your offer conditions, have the scores sent immediately. They will then notify UCAS that the offer is firm, and you must confirm your acceptance in the online UCAS system.
Best of luck! My son decided to apply at the last minute and I was rushing to the post office overnighting everything, so it was quite hectic. Start early!