STAYMAN 

This is probably the most well-known convention (this or Blackwood) in bridge. It is used after a notrump bid to look for an 8-card major suit fit. After partner's 1NT opening, responder will want to look for a 4-4 major suit fit, with say: A Q 6 5
6 5
K J 8 7
J 3 2. To ask opener if he has a 4-card major, he responds 2.

After a 1NT opening (or overcall), 2 asks the 1NT bidder about majors.

The responses to 2 are:

2= No 4-card major
2= 4 (or 5) hearts
2= 4 (or 5) spades

After opener's rebid

Here is the brief overview of what it means if the Stayman bidder takes further action:
On the 2-level, everything is natural and not forcing (example: 1N-2;2-2). I suggest that auction is "pass or correct" -- sometimes referred to as "Garbage" Stayman or "Crawling Stayman. The responder can have only 4-4 in the majors and just wants to stop in 2 ore 2.
On the 3-level, responder's raise is invitational, of course. Other 3-level bids (using LC Standard) are discussed here.
On the 4-level, a raise to game is to play, other bids are splinter bids (example: 1N-2;2-4).
Partnerships wishing to cover most of the details can read about them here.

The Stayman bidder should usually have at least invitational values. The only exceptions would be:

A) A hand that is willing to pass anything opener answers (including 2)

B) A hand that has both majors (at least 4-4) and is willing to pass 2/2 or correct a 2 answer to 2 to show a weak hand with both majors

After a 2NT opener (or non-jump overcall), 3 is Stayman with the same schedule of responses (a level higher).

Puppet Stayman

Some partnerships use this variation of Stayman which uncovers not only 4-card majors, but also 5-card majors. I recommend it only for very experienced players, and only on the 3-level.  I like 2 to be regular Stayman because it allows for various weak responses (with the ability to stop on the 2-level).  In LC Standard, I prefer to use Puppet Stayman only as a jump to 3 after 1NT.  (After a 2NT opening, I don't like Puppet Stayman because it is hard to show 5-4 in the majors). For more on Puppet Stayman, click here.