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.
Universities in Europe
The first university founded in Europe was the one in Bologna, in 1088AD, 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. Note that the age of a university does not necessarily reflect on its status.
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 weren't bad. 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 brawls between local townspeople were not unknown. A successful university career was often a stepping-stone to important positions at Court or in the Church.
Universities in the SCA
In the SCA, a "University" can be a branch of the society and is referred to as a college. A university is often a popular recruitment base for the Shire or Barony in which the College lies. The kingdom of An Tir has the kingdom wide University of Ithra.
In New Zealand the use of the word University is reserved 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.