The University is a place of higher learning and knowledge for your people and is the key to growth for your empire. Other buildings cannot be upgraded higher than the university level plus 1. Your university's level is empire wide where construction level of buildings on other planets is concerned. Many of the more powerful buildings that can be constructed require a very high level University. The University is also a source of happiness for empire.
Note: If you do lose you University for some reason by Demolishing it, Abandoning the planet it is on, or should it be lowered in level because you downgrade it or because of the spy mission Appropriate Technology, your empire will still maintain the same University level of technology that you had before the loss. To increase your University Level you will need to rebuild up to the level that you had before and then continue to level the University 1 level at a time.
It used to be possible that you could take control of a planet with a level 29 University even if you only had a university of for example level 10 and then upgrade the University a single level and suddenly jump from University level 10 to University level 30. This is no longer the case. To increase your University level now from 10 you would need to upgrade a University to level 11.
There is no way to lower the University level of your empire currently it game. It can be done by a game admin but will likely only happen in the case of rules violations.
Once your University level reaches 30 it is currently impossible to increase the level further. Some players like to keep their University at this point because of the happiness it produces despite the high maintenance costs. Others keep it in hopes of eventual higher level tech coming out, or just as a memento of their hard work.
Each university level above 25 will give +1 to the maximum level a normal resource building can go; for a max of +5 at university level 30 (the university itself is not required to maintain this bonus). Added to the maximum stockpile bonus, that means the capitol's normal resource buildings can reach level 30 while those on other planets can reach level 20.
If you look at the Tech Tree you might think to yourself, "Why would I upgrade the University past level 29?" The answer is that you are probably going to want to be able to build your buildings up past level 29. Besides, the University is a great source of happiness for your people. And in the future we'll likely introduce more buildings that will require an even higher university level.