Oxbridge is a term used in the United Kingdom. It is a combination of Oxford and Cambridge University, two elite higher learning institutes found in the British Isles. Cambridge University was founded after a falling out between certain factions of Oxford University.
While discussing his strained relationship with Mycroft, Sherlock mentions that, given their brief romantic relationship, he understands why Watson is defending his brother, and that perhaps she'd be defending him if they had had a sexual liaison. When he conceived the series, one of Robert Doherty's motivations for flipping Watson's gender was to demonstrate that men and women could have a platonic working and living relationship without ending up romantically involved.