If you are talking about the NFL, NHL, NBA and MLB then the answer is the Chicago Cubs who have not won a championship since 1908.
There are a number of teams that have never won a championship, but the Cubs 101 years without a title is the longest drought.
Coincidentally, MLB has been around the longest.
Longest championship drought in NBA: Sacramento Kings(1951, when the team was the Rochester Royals)
Longest championship drought in NHL: Chicago Blackhawks(1960-1961)
Longest championship drought in NFL: Arizona Cardinals(1947, then located in Chicago)
Speaking in terms of the four major American sports(Baseball,Basketball, Football, Hockey), the Chicago Cubs of Major League Baseball have not won a World Series since 1908.
The City of Cleveland has 3 professional sports teams and has not won a Championship since 1964 making them the city that has gone the longest without winning a Championship. The Chicago Cubs of Major League Baseball are in the midst of a 102 year World Series Championship drought. (1908 was their last W.S. Title) The Cubs haven't even appeared in a World Series game since 1945. This is by far the longest individual drought in all of sports.
LSU and FSU
Boston i think
Kansas City Royals
The National Hockey League (NHL), plays for the Stanley Cup championship trophy.
One of the major sports in Kuwait is soccer.
Most states with multiple sports could claim that distinction depending on how far back you go. For instance, Michigan has won championships in all major sports, even football, though you have to go back to the 50's, before the Super Bowl.
The most major sports team in Pennslyvannia is the Pitsburg steelers
UConn has a verity of sports teams for colleges
The major sports that use helmet cams during their broadcasts are typically NFL, NHL and Major League Baseball. You will note that these are all American sports.
Kayaking, skiing, soccer, and tennis are some of Sweden's major sports. other sports are just like ours!