What is Stayman?
In every auction we always ask ourselves whether we have enough points for game and whether we have a major suit fit. When partner opens the bidding with 1NT and we have at least 9 points, we know that we want at least to invite game but, if we have a 4-card major, we have no way of knowing whether partner has 4 cards in our major as well. The Stayman convention solves this problem. When partner OPENS 1NT, you can bid 2C Stayman, which simultaneously tells partner that you have at least 9 points, that you have at least one 4-card major, and that you want to know if the 1NT opener has a 4-card major as well.
A Note on 8 VS. 9:
You may have heard that Stayman can be done with only 8 points. This comes from three places. First, it comes from the days when 1NT used to be 16-18. Second, some teachers teach 8 because many good eights are really nine but they just don't want to deal with teaching what a good 8 is verses a bad 8 to their newer students. A good 8 is an 8HCP hand that really evaluates to 9 because it has either a five-card suit that allows you to add one point for the fifth card or because it has two strong 4-card suits that have a good chance of the fourth cards in those suits being a trick. Third, some teachers teach to only accept a game invite as opener with 17, in which case you will have 25 or 26 points. We teach to accept with a good 16 or 17 points in which case you will have 25 or 26 points when responder has 9, and you need at least 25 to make game. In these notes we use 9 for consistency but know that we are referring to 9 total points which could be 8 HCP plus 1 point for a 5-card suit or two good 4-card suits.