A complete answer would use up more time and space than is reasonable. But here are some high-lights:
In Europe, some old universities such as Oxford, Sorbonne, and Bologna developed slowly into being universities and so do not have strictly speaking founders. Others such as Cambridge spawned from existing universities. Universities developed quickly into a necessity for the various states, which meant that rulers actively went about founding them. One important aspect of university education was the education of clergy, so bishops often had a hand in a university coming about.
In the renaissance, universities already have become important institutions controlling effectively access to certain professions. Rulers sometimes asked other institutions, for example, Catholic orders (e.g. the Jesuits) to found universities (e.g. Ingolstadt, Universidad National de San Marcos in Lima) or help in the founding.
In the USA, many old universities developed from Protestant seminaries (Harvard, Yale) or were founded by Catholic orders (Georgetown, Santa Clara) on behalf of the Catholic church.
The vast majority of universities in Europe were founded directly by the state, even if sometimes they bear the name of someone else. In the USA, there are the land-grant university which were founded by the states and are usually the most important local universities, especially outside the Eastern Seaboard. However, founding by religious groups has continued, often because of the desire to offer an education that is not secular.
In Latin America, each country has its peculiarities. Uruguay insisted for a long time (until the end of the dictatorship) that there should be one secular university, the University of the Republic. ORT university was established to help the Jewish refugees flooding into the country in 1947, but had to wait to become a university. The Jesuits were tasked with establishing the Catholic University at the end of the dictatorship because the education at the University of the Republic was seen to be hostile to Catholic students. In other countries such as Chile, you find two universities, the Pontifical and the University of Chile, to which now many other universities have been added, some by religious orders, some by rich people that want to benefit their community or state (such as Stanford and Soros).
You can find similar things happening in other parts of the world.
While universities mostly are not-for-profit, this has changed in some countries, especially the US. A for-profit university is not necessarily of secondary quality but offers a clearly commercial proposition to its students that is advantageous to both, but in the US there are many of shaky quality.
So, most universities are founded by state action and then by religious communities. There are single founders who want to benefit their community (and maybe honor themselves or a loved one.) There are for-profit universities that can have quite a number of students. I cannot think of any university founded by a single academic. Please remember that universities develop and that a small founding of a college that is more like a modern high school can develop in a world-class university over time.
