Stayman and Continuations
The Stayman convention is used to find a 4-4 major fit after a 1NT or 2NT opening.
With two balanced hands and 25+ points, 3NT is often the best contract - but not always! If your side has an 8 card heart fit or an 8 card spade fit then 4♥ or 4♠ might be better. After a 1NT opening, there is no way that responder can bid naturally to show a four card suit because if responder bids a suit at the 2 or 3 level that shows a 5 card suit.
Instead of using a 2♣ response to 1NT to show clubs, you can change the meaning of 2♣ to Stayman, asking if partner has a 4 card major.
Stayman 1NT - 2♣ = Stayman, do you have a 4 card major? -- 2♦ = I do not have a 4 card major -- 2♥ - I have a 4 card heart suit and I might have 4 spades -- 2♠ - I have a 4 card spade suit
In the hand below, North and South are playing Acol so the 1NT opening shows 12-14 points.
South has enough points to bid game but 3NT will go down because the opposition can take four heart tricks and a diamond. However, 4♠ will make with careful play.
This time the opener has denied a 4 card major.
Responder simply bids 3NT now, knowing there is no major fit.
Opener shows a 4 card heart suit but that's not the major suit that responder was looking for.
Responder bids 3NT.
In these next two examples, North and South are playing a 15-17 1NT opening.
Opener shows a 4 card heart suit but that wasn't the suit partner was looking for. 2NT is still inviting to game, though, and shows the same number of points as a direct 1NT - 2NT bid would have shown. North is happy to accept the invitation.
This time North and South find a spade fit.
Responder's 3♠ invites opener to bid game with a maximum hand.
South doesn't fancy playing in 1NT
South will pass no matter what the opener bids.
Stayman Continuations
Stayman Continuations 1N - 2♣ -- 2♦ --- 2♥ = 4♠+5♥, to play ( or Crawling Stayman ) --- 2♠ = 5♠+4♥, to play --- 3♥ = 5♠+4♥, GF (Smolen) --- 3♠ = 4♠+5♥, GF (Smolen)