Complex sentence: The 1999 team brought Gonzaga basketball to national prominence with an impressive run in NCAA tournament, advancing to the West Regional final (Elite Eight), and taking the eventual national champions UConn down to the last minute.
Split into simpler sentences: The 1999 team brought Gonzaga basketball to national prominence with an impressive run in NCAA tournament. The "Zags" advanced to the West Regional final (Elite Eight), taking the eventual national champion Connecticut down to the last minute.