The phrase "big dance" is another term for the NCAA Division I Basketball Tournament. Every year, 64 teams in Division I Basketball qualify for "The Big Dance," all playing for a chance to be crowned the top team in the country. A season where you do not make the Dance is considered a failure by the standard of many schools.
The name came about in 1977 when Marquette's head basketball coach was questioned about this eccentric blazer he wore frequently during the season. When asked if he would wear the suit jacket during the NCAA Tournament, the coach replied "You gotta wear the blue blazer to the big dance." Marquette would go on to win the championship shortly after, and the term "Big Dance" has stuck ever since. Amazing how a joking response from a basketball coach went on to be how one of the biggest basketball event's nickname was coined.