Lesson 1: Stayman

After partner’s 1NT (or 2NT) opening, there might be a major suit fit. Playing in a major will usually work better if there's a shortage. You might lose control in no trumps if the opponents lead the shortage, but if it's a suit contract, you'll be able to trump to keep control. 

Use the Stayman Convention to check for this. Bid 2♣ (Stayman) after partner opens 1NT (or 3♣ after 2NT) to ask opener to bid a four-card (or longer) major. Opener bids the major they hold, starting with hearts. With no four-card major, opener bids 2♦.
Partner opens 1NT and it’s your call with this hand: ♠KJ94 ♥K985 ♦83 ♣A74. Bid 2♣, asking about opener's majors. You have 11 high card points and know the partnership belongs in game. The only question is: which game? 3NT, 4♥, or 4♠?  
After a 2NT Opening Bid
Stayman may be used after an opening bid of 2NT. Partner opens 2NT (20,21 points) and you hold: ♠T742 ♥K85 ♦T842 ♣A7 Respond 3♣, Stayman, asking opener to bid a four-card major. You want to be in game. If the partnership uses a strong 2♣ opening, Stayman is also used after 2♣ p 2♦ p  2NT (balanced 22-24 pts), 3♣ would still be Stayman.

Stayman: Summary
After 1NT (or 2NT), a response of 2♣ (or 3♣) is Stayman, asking opener to show a four-card (or longer) major. Stayman is used for hands with four cards in one or other major. Opener responds as follows:
2♦ No four-card or longer major suit.
2♥ A four-card or longer heart suit. With both majors, opener rebids 2♥ first.
2♠ A four-card or longer spade suit (not four hearts).