All Cohanim and Levi'im are descended from the tribe of Levi. During the time of the temple, a Cohen would have to provide documentary evidence of their ancestry in order to serve in the temple (כהן מיוחס). Since the Babylonian exile, all such documentary evidence has been lost (with the exception of perhaps the Rappaport family). Some interesting genetic studies have also been done relating to the shared ancestry of Cohanim. Today halachically, anyone who claims to have a tradition as being a Cohen or Levi is accepted as one, for the sake of the privileges such a role entails (like getting the first or second aliyah, being able to give the priestly blessing).
Aside from Cohanim and Levi'im, there are many people who have family traditions that relate them back to Kind David (placing them in the tribe of Judah). These usually consist of direct family trees going back to a Jewish sage (rishon or achron) who lived some time in the last thousand years, and who have been related back to King David (for example, Rashi and the others).
Aside from that, no one can claim with even a small degree of certainty to which tribe they are descended from (assuming patrilineal descent). After the exile of the "10 Lost Tribes" into the Assyrian empire, there were either 3 or 4 tribes left - Levi, Judah, Binyamin and probably Shimon. Most today are descended from one of the last three (with the greatest likelihood of descent from Judah, as it had the biggest population at the time of the first exile).