The word 'University' is derived from the Latin word for guild. The church established schools with the purpose of educating people in a particular profession (a non-trade profession), and then universities arose purely for higher education. In the Middle Ages, it was mostly churches or courts who had the better book collections of the times, so therefore those involved with these institutions (who were mostly those with lots of money) therefore had access to higher education. The first university, Al-Azhar, was founded in Cairo, Egypt in 970BC, which was followed up with more being founded predominently in Italy, France, England and Spain (during the 1100s) and later in Germany and Austria (1300s). They had various faculties, as they do today.
Anton notes : The premiere theology university of medieval Europe was the University of Paris at Notre Dame, but Bologna was the place to go for Law, and Palermo for Medicine.
In England, Oxford and Cambridge werent bad, at least until they got purged of Catholics in the Reformation.
Most medieval universities, and students at them, were supported by the Catholic Church, or by bequests from the rich and the pious.
University students had a fair amount of control over the faculty, and town/gown brawls were not unknown. Royal troops were never used to restore order to Notre Dame. Honest.
A successful university career was often a stepping-stone to important positions at Court or in the Church.
I dont know about Al'Azhar being established at 970 BC ... I suspect it was formed in 970 AD ...
In the SCA, a University can be a branch of the society and is referred to as a college. Universities are popular recruitment bases for the Shire/Barony in which the College lies. An Tir has the kingdom wide University of Ithra.
Note that the use of the word University is reserved in New Zealand for actual government acredited institutions approved to use that name, and therefore the Kingdom of Lochac does not use the word University to describe any of its internal institutions.