If you plan arriving Tartu via Tallinn Airport, you should take the direct bus to Tartu (operated by Lux Express). The travel to Tartu takes approximately 2 hours 20 minutes. The final bus stop is the Tartu Central Bus Station at the center of Tartu.
It is advisable to buy the bus ticket in advance via internet (at T-pilet) or from the ticket machine on the ground level (0-level, one floor below baggage claim) of Tallinn Airport. Note that Tallinn Airport is called Tallinna lennujaam in Estonian. If you have bought your ticket via internet, you have to print the ticket out yourself. You can do this at your home or at Tallinn Airport. The ticket machine can print your ticket free of charge (you have to know the ticket number). Also, it is possible to print your ticket at the Information Desk, but this costs 1 EUR.
Buying the ticket in advance is especially important, if you will arrive on Sunday evening — many students are coming after weekend back to Tartu and there might be no seats available.
It is also possible to start the bus trip to Tartu from the Tallinn Central Bus Station (Tallinna Bussijaam). You can easily get from airport to the Bus Station by taxi or by bus number 2.