In Civilization and Civilization II, the University is a City Improvement related to knowledge production, boosting the knowledge production in the city it is built in by 50%. The improvement requires the university knowledge advance, and a good amount of production squares. It also requires that a library already be built in that city.

Everything that could be said about a library can also be said about a university, that is, that any city that has the resources to build a library, and will benefit from the bonus, perhaps doesn't need it all that much. Since a university is somewhat expensive to maintain, a city must already be producing quite a bit of knowledge for the benefit to be worthwhile.

I find myself mostly building universities fairly late in the game, and then more for the long term goal of getting Future Tech advances for score then to win in the knowledge race against other Civilizations. If I do build universities early on, it is because I have a small amount of large cities, or have switched to an advanced form of government early on.