No, President Barack Obama cannot run for the presidency of the United States in the next election (due to take place on November 8th, 2016).
Why Can't Obama Serve a Third Term?
For many years, two terms was an unofficial limit, but there wasn't a law as such preventing someone from seeking a third term in office. Franklin D Roosevelt actually served four terms as President, with the third and fourth due to exceptional circumstances (the US entering the Second World War), although Roosevelt died before he could serve out the fourth term in full.
It wasn't until 1951 when the twenty-second amendment to the US constitution was ratified that it became legally impossible for a president to seek a third term.
This film explains more about the twenty-second amendment: