The most common option in Estonia is to rent your accommodation, either in the private sector or if you are connected to universities, then rent a dorm room.
If you are here for only few years, renting is probably better option than buying a property, since the latter also entails quite few obligations.
Since a number of students land in Tartu every year, demand sometimes outweighs supply and prices in Tartu can be somewhat higher than in other cities, but here´s some background.
The types of accommodation in Tartu vary widely, from old Soviet-style apartment buildings to large houses with gardens. Both furnished and unfurnished accommodation is available. However, some criteria are rare to find: a toilet with bidet; an apartment with several bathrooms, air condition (but the weather in Estonia almost never requires any air conditioning!)
The most common option in the Estonia is to rent a private sector house. Renting this type of accommodation is not subsidised. If you are not an EU-citizen, you have to have a residence permit to make a rental agreement.
To find a rental property, you can
Take up the challenge and search yourself, eg through social media groups or specialized websites (the main keywords would often be in English)
Contact a real estate agent (broker)
The options for rent are normally either partly furnished or unfurnished. There is also different type of heating - from central heating to firewood-heated owens.