There are several kinds of accommodation alternatives in Hungary but the cheapest one is usually a dormitory belonging to the students’ homes section of the university you attend. One advantage is that it is cheap, you will always be surrounded by friends and great communities are often created by living together.
On the other hand, renting an apartment is also a feasible option if you prefer privacy. This way, you can get your personal insight of how Hungarians live. Cost can be very different depending on living in or outside of the capital: HUF 70,000–150,000 or HUF 120,000–200,000 in Budapest a month plus overheads. The landlords usually require you to pay a deposit in advance, which usually equals a minimum of 2 months’ rental fee (this will be paid back when you move out.)
If you need help, contact your international co-ordinator or mentor at your university, who will guide you in the right direction. There are usually dedicated Facebook pages. Some students find agencies useful in the process.
Wherever you live, the most important thing is to find a place where you can study comfortably and it is a plus if you can share it with your new friends.