The Jacoby 2NT conventional raise was invented by Oswald Jacoby (who also invented Jacoby Transfers). To use this convention, your partnership agrees that after you open 1 or 1 and your partner is an unpassed hand, his jump to 2NT shows:
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 1 or 1 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 2N is natural, showing a balanced hand of 11-12 points.
If you are playing Splinter Bids, then you Splinter to show a game force with 4+ trumps and a singleton or void, and use Jacoby 2NT for hands with no singletons or voids.
If you open 1 of a major and partner responds 2N, the first responsibility is to show a singleton or void, no matter what your strength:
If you have extra values, but no singleton or void, describe your strength as follows:
For example, after you open 1 and partner responds 2N, the meanings of your rebids are:
This theme of jumping to game with a weak hand and bidding smaller and smaller steps with stronger and stronger hands is repeated in many conventions, so it's worthwhile remembering that after an asking bid, a jump to game (called "fast arrival") is virtually always a sign-off while the smaller the increment of the next bid, the stronger the hand.
(Source for the above specifications: Bridge Conventions Complete by Amalya Kearse. Other meanings of rebids by opener can be found in other sources.)