Did they have entrance tests? Were certain groups officially banned from entering? How much did one have to pay? Did they have scholarships for talented studends? How these criteria differed between countries and over time?
More details can be found in the related Wikipedia article.
University students typically had one of three sponsors:
The admissions criteria and payments were set by the respective sponsors.
That is the church and crown had their own "feeder" schools, and chose the best students of these to take university degrees, and generally paid for the students' education. These choices were made on the basis of public policy.
Wealthy families might "home school" their children, perhaps hiring tutors affiliated with universities, then sending those children to whichever universities would accept them, with the family paying the way.