Oxbridge is an umbrella term for the universities of Oxford and Cambridge, which have very similar traditions and admissions procedures. The universities are the two oldest in the UK (with Oxford teaching since 1096 - making it the oldest university in the English-speaking world, and Cambridge established in 1209). They are extremely well renowned (ranking second and third in the most recent Times Higher Education world 'top 200 Universities' assessment).
Applicants for Undergraduate degrees can only apply to only one of the two in any application cycle, and for only one course, (unless applying for an Organ Scholarship, or for a Second Undergraduate Degree, in which case they may apply to both universities but with the course restriction remaining), and both impose an earlier deadline than is common for other UCAS applications (usually around mid-October). Postgraduates can apply to both simultaneously.
Both universities are each made up of self-governing colleges (31 at Cambridge, 39 at Oxford in addition to seven Permanent Private Halls), and undergraduates apply directly to, and are admitted by, the colleges themselves (through the central Undergraduate Admissions Office), rather than to the departments as is common amongst most other universities. (At the postgraduate level, however, admission is by the departments first and foremost.)
Read the 'Applying to Oxbridge' for more details on applying there.