When you are first to act pre-flop in Hold'em or Omaha you are said to be under the gun. You sit immediately to the left of the big blind and must act first after the hole cards are dealt. Your position is very precarious hence the ominous sound of the term.
A number of Hold'em terms originate from the games underground and dangerous roots out West. A few "violent" examples include: drawing dead, gut-shot draw, and of course under the gun. You must make the initial bet out of position and you have the least amount of time to determine your course of action. So the pressure you feel is akin to being under the gun!
