There are several kinds of accommodation alternatives in Hungary.
The cheapest one should be a dormitory belonging to the students’ homes section of the university you attend. One advantage is that it is cheap (HUF 10 000- HUF 30 000/month), it is an easy way of making friends and of establishing a social community fast. If you decide to go with this more affordable type of housing, then you should act fast because the dormitory places are limited and vary from university to university. Check the apllication deadlines and methods before your arrival not to miss it.
The are also so called private dormitories in some Hungarian cities ( around HUF 40 000 - HUF 50 000/ month) which do not belong to a university or a college. These are not so cheap as the institutional ones, but you can still spare some money through this way compared to private flats or rooms. It can be a plan B.
On the other hand, renting an apartment or a room in flat is a more feasible option if you prefer privacy. The costs depend much more on the fact in which city you are renting, there can be big differences between fees when it comes to Hungarian cities. The average cost is about HUF 100 000/ flat, and about HUF 50 000- HUF 70 000/room.