Jacoby 2NT -- Forcing Major-Suit Raise

Playing a Standard American bidding system, you have three direct ways to show trump support after partner opens 1H or 1S:

A simple raise to two (1S-2S), to show 6-9 pts.

A jump to three (1S-3S), which can be played either of two ways, depending on your preference: as a forcing raise (12+ pts.), or as an invitational limit raise (10-11 pts.).

A jump to game (1S-4S), to show limited high-card points (usually about 7-10), but great trump length and playing strength.

If you play the more modern system of using the jump to 3 as a limit raise, you have a somewhat awkward problem when you hold a forcing-to-game hand with trump support. One solution is to bid a new suit first, temporarily "concealing" your support, then jump to game in partner's suit at your next turn (an auction like 1S by partner - 2C by you - 2NT - 4S).
There are a number of different conventions designed to help you describe these strong hands. One of the most popular is the Jacoby 2NT, which was invented by Oswald Jacoby (who also invented Jacoby Transfers). To use this convention, your partnership agrees that after you open 1H or 1S and your partner is an unpassed hand, his jump to 2NT shows:

Forcing-to-game strength (12+ points)

Good trump support (4 or more cards)

The 2NT response forces you to at least game level. It is not a suggestion to play in notrump; it shows support and asks the 1H or 1S opener to further describe his hand.
Note that the Jacoby 2NT convention is "on" only if responder has not previously passed in the auction. If you are a passed hand, your jump to 2NT is natural, showing a balanced hand of 11-12 points.
　
Opener's rebids

In addition to giving responder a direct way to show a forcing hand, the Jacoby 2NT improves the accuracy of your slam bidding. Its biggest advantage is that it gives opener room to further describe his hand. There are a number of different meanings you can assign to opener's rebids. Here's one of the most commonly used structures:
If you open 1 of a major and partner responds 2NT:

Your first obligation is to bid 3 of a suit where you have a singleton or void.

If you have a another semi-strong 5-card suit, show it by jumping to 4 of the suit.

If you have extra values (but no singleton or void), describe your strength and pattern by rebidding 3NT with a fairly balanced hand or 3 of your suit with extra length (a 6+-card suit).

If you have a bare-minimum opener with no singleton or void, jump to 4 of your suit. This is your weakest rebid. It shows a fairly balanced hand with no interest in slam.

For example, after you open 1S and partner responds 2NT, the meanings of your rebids are:

3C, 3D or 3H = Singleton or void in that suit.

3S = 14+ pts. with extra length in your suit (or 19+ pts. if you're balanced), no singleton.

3NT = 15-18 pts., balanced, no singleton.

4C, 4D, 4H = Good 5-card suit (QJxxx or better).

4S = Minimum hand (12-14 pts.), no singleton.

Once responder has this information, he can decide how to proceed. If he's not interested in a slam, he can just jump to game. If he wants to explore further, he can cuebid a new suit to show an ace. This requests opener to cuebid an ace.

The Jacoby 2NT in action
(1)   KJ832        A10764
      A65          KQ5
      10           872
      A954         K7


     Opener     Responder  
       1S           2NT
       3D           4NT 
       5H           6S
After responder uses the Jacoby 2NT, opener bids 3D to show his singleton. This is encouraging news for responder because it means his club and heart honors will fit well in partner's hand. Knowing that his xxx suit is covered by opener's singleton -- and that opener's outside strength is in clubs and/or hearts -- responder can use Blackwood to check on aces, and then bid his 24-point slam.
Note that responder would evaluate his hand very differently if opener's answer to the Jacoby 2NT had been 3C or 3H. Singletons in these suits wouldn't be helpful because responder's kings may well be "wasted" cards opposite the singleton. Over opener's 3C rebid, responder would just jump to 4S to show that he no longer had interest in slam.
(2)   QJ10543      AK72
      AKJ          105
      32           KQ54
      K9           A62

      Opener      Responder
        1S            2NT 
        3S            4C 
        4H            6S
Opener's 3S rebid denies a singleton, but promises a good (somewhat distributional) opener. Responder shows slam interest by cuebidding his club ace and opener cooperates by cuebidding 4H. This shows the heart ace and denies the diamond ace (since he'll cuebid his cheapest ace). Responder now knows that the partnership has extra values, control of every suit and exactly three aces. Since he doesn't need to use Blackwood, he jumps to 6S.

　