The 3344 convention is a simple version of Eric Rodwell's "Game Try Convention".
When your partner opens a minor and you respond a major and they raise to the two level, they do not always have four card support for you.
They can have a few different hand types which would make a raise with 3 card support more palatable than bidding 1NT.
With an invitational or even a slam going hand how do you find out?
You might not be sure of the whether you want to play in game in your major or whether 3NT is better.
The answer is to use the 3344 convention by bidding 2NT.
You must have at least an invitational hand to use this convention.
The responses are:
There is another version of this of course which is suggested by Mel Colchamiro in a Bridge Bulletin article which uses 3434 responses and which he calls Spiral.
The responses for his version are:
Which ever one you choose make sure your partner is playing the same one.