There are plenty of places to choose from in Szeged (from student hostels and small guesthouses to exclusive hotels).
For a budget accommodation, see Romzsa Tódor Kollégium, which is a student hostel located in the centre of Szeged, facing a nice park with a tiny church. It offers rooms for 2 to 5 people, free Wi-Fi and free parking on-site. Rooms are bookable online. For other hostels, see websites below.
Familia Guesthouse offers single rooms for 28 euros (with breakfast for an additional 5.2 euros), but also has rooms for 2, 3, or 4 persons. It is located near the Franciscan Church, just halfway between the railway station and the Faculty of Arts building of the University of Szeged, and is within easy reach (10 minutes walk) from downtown as well. You can book rooms through their own website. Other guesthouses with a good location and a friendly price include Mátyás Panzió, Szivárvány Panzió, Mosoly Apartman, Kata Panzió, and Antik Apartman. For booking, see websites below.
Mátrix Hotel and City Hotel Szeged have relatively decent prices, while more exclusive service is provided by Hotel Korona, Dóm Hotel, Novotel Szeged, and Tisza Hotel.
All of the above and several others are bookable through the following sites:
http://hungary.accommodationbook.com/Szeged (in English, offering probably the largest selection)
http://www.hellotourist.net/index.php?nyelv=3 (in English, German, and French)
http://www.booking.com/ (in English, German, and French)
See map including most of the above possibilities.