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 Conventional response to notrump to ask about majors 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" Conventional response to notrump to ask about majors Stayman or "Crawling Stayman. The responder can have only 4-4 in the majors and just wants to stop in 2 or 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 An unusual jump showing 0-1 in that suit and support for partner splinter bids (example: 1N-2;2-4).
Partnerships wishing to cover most of the details can read about them here.

The Conventional response to notrump to ask about majors 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 Conventional response to notrump to ask about majors Stayman with the same schedule of responses (a level higher).

Puppet Stayman

Some partnerships use this variation of Conventional response to notrump to ask about majors 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 Conventional response to notrump to ask about majors Stayman because it allows for various weak responses (with the ability to stop on the 2-level). In LC Standard, I prefer to use An artificial response asking opener for a 5-card major Puppet Conventional response to notrump to ask about majors Stayman only as a jump to 3 after 1NT. (After a 2NT opening, I don't like An artificial response asking opener for a 5-card major Puppet Conventional response to notrump to ask about majors Stayman because it is hard to show 5-4 in the majors). For more on Puppet Stayman, click here.