Out of the 68 teams that begin in the March Madness Tournament, only 16 teams qualify for the Sweet 16. To make it there, teams have to win in the Round of 64 and the Round of 32. If a team played in the First Four, they also have to win that game to even make it to the Round of 64.
Before all that, teams of course have to qualify for the tournament!
The Sweet 16 is the regional semifinals. Within each region there are four teams left at the time of the Sweet 16. Teams which are victorious in the Sweet 16 advance to the Elite 8, which is the regional championships.
A Cinderella story occurs when a team which is ranked 11th or worse in the region advances to the Sweet 16 or further. There is usually one Cinderella story each year, and it will likely get a lot of media coverage.