Usually, not very much! Some enquirers find this surprising but just consider how much (or how little) information your school holds about you. The first port of call for information on anyone who was up at Oxford between 1500 and 1892 is Joseph Foster’s Alumni Oxonienses which gives brief details, compiled from the University’s registers, of all matriculated members of the University. Most large libraries should have a copy. The companion volumes for Cambridge are Venn’s Alumni Cantabrigienses.
It is most unlikely that you will find any details of mothers, wives, or sisters. Women didn’t feature very strongly in pre-20th century Oxford! Try the International Genealogical Index on-line (see Links). It will also be unusual to find dates of birth, essays or dissertations, or examination results.
If your ancestor was famous, try the Oxford Dictionary of National Biography or even Who was Who. The vast majority of Oxford graduates, until the 20th century, became clergymen so it may be worth looking at the published Clergy Lists and their successor volumes, Crockfords, as the database of Church of England Clergy (see Links).
Depending on the period your ancestor was alive, college archives should be able to confirm the information in Foster and, possibly, add some information about their career in obituary in a college magazine. college and their first posting on going down. For more recent men, you may find an obituary in a college magazine or published in a newspaper. If your ancestor was a senior member of the college, there may be a portrait. These are described in Mrs Lane-Poole’s Portraits in the Colleges and Halls (1926). Some colleges have their own printed registers of alumni. The archivists are bound by the terms of the Data Protection Act so we are not allowed to give out information on living members without their permission, and we must show discretion about the records of recently deceased alumni.