Most of the apartments come to the market at least partially furnished. Means: there will be at least kitchen and bathroom fitted with all home appliances. Rooms might come furnished or unfurnished. This is always mentioned in the advertisement text, but sometimes photos might be confusing (furniture on the photos might belong to previous tenant and s/he is going to take it with him) - so always check the text, and ask from the owner or realtor (that is called "maakler" in Estonia).
I would recommend to look for apartment directly from the owner, to avoid paying extra 1 month fee as realtor's fee. Beware, that price might be different for winter and summer months, as typically it is based on fixed price you pay to the owner + utilities that depends on the season, temperature outside and your consumption (electricity, hot/cold water, internet/TV package, sometimes even parking lot). I would even count for at least 500 EUR fixed rental fee to the owner plus up to 250 EUR for utilities per winter month, or 150 EUR per summer month, for very good quality fully furnished 2 room apartment in the city center.