There are many hotels and hostels in Tartu, especially it being an "intellectual and cultural hub" with students coming from all across Baltic States. We recommend searching for accomodations with booking.com.
We can't stress this enough: book a room as early as possible. Maybe right now. Because it's summer, and Tartu is quite popular with tourists and students hoping to be admitted to the University.
A member of OSM EE Jaak Laineste has some suggestions for you. Those accomodations are marked in yellow on the map.
All hostels within about 15 minutes walk to the city center, from best to worst by personal preferences:
Tip: some smaller villa-apartement places (e.g. Vikerkaare) may have no 24-hour staff, so you need to call ahead to arrange arrival on late hours. Some otherwise cheap places are on the edge of city (some 5-6 km from center) and I'm not that sure about public transportation there. The only camping place near the city center would be in my backyard - free lawn provided.