"To drop the ball" is an idiomatic expression in American English. Someone who "drops the ball" has made a stupid mistake, often one that lets others down. The expression usually carries the implication that the person being described has made a mistake through carelessness, inattention or incompetence.
The origins of this expression come from sports, although exactly which sport is unclear. There are several sports in which dropping a ball that's in play can cause negative consequences for a player's team. In baseball, for example, a fielder who drops a ball will be unable to throw it back quickly enough to catch a runner out. Similarly, in football, dropping the ball makes it possible for the opposing team to take possession. Both of these situations lend themselves to the analogy in the expression "drop the ball," that one player's careless error has created difficulties for the entire team.
This phrase entered colloquial American English in the mid-20th century. By the 1940s, it was a common expression indicating a careless error. During this period, a secondary sense developed, in which "to drop the ball" meant not only to make an error but specifically to miss some kind of opportunity. It appears to be rarer in British English, which might tend to support the theory that it originated in either baseball or football. The phrase's connotation of missing an opportunity suggests that the expression refers to a player jumping to make a catch, rather than a ball carrier in football, who holds the ball for a longer period.
Like other such expressions, "drop the ball" can occur equally in accusations and apologies. A teammate or superior might accuse someone of "dropping the ball" and letting others down. Similarly, it is not uncommon to hear a person who has made a mistake acknowledge it by saying "I really dropped the ball on this one."
One alternative use of "drop the ball" is connected to New Year's Eve. In New York City, the new year is marked by a lit ball which descends down a mast atop One Times Square, counting down the last few seconds of the year. This ball is sometimes said to be dropped, but the two expressions are not otherwise related. Similarly, the British English expression "to balls something up" has a similar meaning, being used to describe someone who makes a mistake, but relies on an alternative meaning of "ball."