A university is an institution of post-secondary education, usually meaning one that has many colleges under its auspices. In the United States, there is, however, no rule that this must be the case for a school to use the word university in its name (e.g. the three main service academies).
Other countries may have legal or academic requirements for a school to do this. A case in point is the United Kingdom, which used to have degree-issuing institutions known as polytechnics teaching subjects of a more vocational nature, which were upgraded to university status through the Further and Higher Education Act in 1992. Hong Kong did the same to its polytechnics in 1994 and 1995 and to the then Shue Yan College, now Hong Kong Shue Yan University, in 2006. In these cases, the change in status allowed them to validate their own degrees rather than have them validated by an outside body such as the Council for National Academic Awards in the UK.
True universities will often not only have many distinct undergraduate colleges, but usually several prominent professional or graduate schools as they are known in the United States. Graduate schools offer advanced degrees in many academic fields, and sometimes also offer professional degrees in fields like Medicine and Law.
The most prominent universities are the older colleges that have grown over the centuries since their founding, and (in the United States) the land grant State universities which were founded to make higher education affordable for the common man (and, eventually, woman). Many of the most prominent private colleges and universities were created to train pastors and other religious leaders and, even without that mission, they were closely affiliated with a church or denomination; the majority have since either severed their ties with that church/denomination (and thus have become private secular universities) or that church/denomination has become liberal (often as a result of the university's liberal bias). Land grant institutions, on the other hand, usually were founded with economic development and general education as their primary missions (usually agriculture and mechanical arts, the latter a predecessor to modern engineering) and were founded as government-supported, secular institutions; today they have largely expanded beyond their historic base to include other courses.
Another group of universities, largely second-tier behind the prominent ones, were started as "normal schools" which focused on training teachers in educational philosophy. These universities still perform that mission but have also expanded to include other courses, usually those focused on local or regional needs (e.g. Stephen F. Austin State University, located in the forested area of East Texas' Piney Woods, has a significant forestry program).