Kaarsild is the arch-shaped pedestrian bridge close to the main square. The legend says that unless you walk over the arch (not the bridge, but the arch) you will not be accepted as a student at the Tartu University. By the way, nowadays the police unfortunately gives fines for these tests of character.
Most probably you'll end up crossing it anyway, because there's much to see on the other side. So why not do it in a proper way? I mean there are stories of people sleeping or even having sex on the bridge...
Those afraid of the police (or heights) can enjoy some of the random artwork they sometimes exhibit on the bridge.