This is a poker term used to describe a specific position at the table in fixed position games (most notably Texas Hold 'Em, Omaha Hold 'Em and their variations). The person under the gun is required to act first after the blinds. This means they must immediately decide to fold, call or raise before any of the other players.
Under the gun is relatively the worst position to be in and one should have a premium starting hand if one plays under the gun at a full table. David Sklanksy wrote a good book that provides good instruction on what kinds of starting hands are acceptable in this position. A usage example:
Recently, while playing Texas Hold-Em at the Ameristar Casino in Kansas City, I had pocket rockets under the gun and I called the $20 bet. Seven others called the $20 and the button raised. I re-raised, everyone called, and the button capped it. The flop was all rags and I capped the betting again with 5 players still in. The turn and river brought no improvement for anyone and let me scoop a pot over $1000.
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