If you live in Poland for more than 3 months, you need to apply for a residence permit, within the first 3 months of your stay.
 For EU Citizens
You need to fill out the document "zarejestrowanie_pobytu.pdf" (Wniosek o zarejestrowanie pobytu obywatela UE) from the following webpage:
Also, you need 3 copies of the original. Furthermore, bring your passport (and 3 copies of it), and your contract with the university (and a copy).
Contrary to what is mentioned on some webpages, I didn't need fotos or proof of health insurance.
If you need to register family members, you will need additional documents. But I didn't do this, so I don't know the exact details.
With these documents, go to ul. Dluga 5 and (if you are an EU-citizen) get a number with an "A" from the receptionist and go to the left corridor on the first floor (room 42). The people in the office speak English.
They will ask you when you entered the country, so don't forget the 3-months deadline.
At the office, they told me that I needed an additional document confirming that I live at the address I gave on the first document. The Polish name is "Potwierdzenie Zameldowania" and I had to go with my landlord to the "Urzad Dzielnicy" (the district office). We had to fill out the document (in Polish with Polish explanations) and my landlord needed to provide a proof of ownership of the apartment. This second document needs to be faxed to the office at Dluga 5 as soon as possible. A month later, I could pick up the permit.
It might be better to get the "Potwierdzenie Zameldowania" before you go to Dluga 5, but it works the other way too.
Unfortunately, my "Potwierdzenie Zameldowania" is a temporary one. So, I have to go again to the district office and get a permanent one. I don't know whether you can get a permanent one on your first visit there, without the residence permit.