Definition of Under the gun
Under the gun (often spelled UTG) is a position on the poker table. Whether you are at a shorthanded table or a full ring table, the player to talk first preflop will always be called "under the gun". There is no official translation of this term, but some people will refer to the position as "under the gun".
Implication of playing under the gun
Since the player in the under the gun position will be talking before all the players, without having any information about what they will be doing, he has to play a smaller range of hands than if he were on the button for example. On the other hand, since he is playing under the gun, the players at the table will give him much more credit than if he were on the button (which can raise almost any two cards). On the one hand, when you're under the gun, you'll have to play a range of stronger than average hands, and on the other hand, you'll have a lot of credit in the eyes of your opponents, which will allow you to bluff with good efficiency.