The term 'Under the gun' (UTG) simply refers to any time you are first to act pre flop in a game with blinds. It has a long history in terms of use of its terminology and general theory about how to act from it, and understanding both its advantages and drawbacks is crucial to playing well from early position.
Some players literally only open the top few percent of hands from under the gun, and this can make your life much easier in deciding what to do with a marginal hand.
The key problem with playing under the gun is that you have every single other player at the table to act behind you. This means that the likelihood of there being a strong hand behind you drastically increases, and this usually suggests that you need a very strong hand to consider raising. However, as the general consensus started to become well known that you need to be extremely tight from under the gun, smart players realised two key features of this position which can actually make it extremely powerful - position later in the hand and perceived strength.
As the majority of players began to realise that most people were not opening very loose from under the gun, they drastically tightened up their own ranges against under the gun raises, thus making the number of callers of under the gun raisers much lower (perfect for a bit of high risk stealing to the astute player). Secondly, although under the gun is first to act pre flop, if the only callers you get are from the blinds, you actually have position for the entire rest of the hand.
This is a huge combination - position, perceived strength, a raised pot, and few callers is what poker dreams are made of, and as a result many players have loosened up their opening ranges from under the gun considerably. This has made it one of the most important positions to observe players closely from.
Some players literally only open the top few percent of hands from under the gun, and this can make your life much easier in deciding what to do with a marginal hand. Conversely, some players are actually opening quite a wide range of hands from this position - again with large consequences on the hand ranges you can put them on.
Being under the gun is one of the most interesting and high risk places to be in blind based poker. It can be a huge weapon, but one that you can easily end up killing yourself with if you don't know how to use it. The usual key elements of poker - observation, solid betting sizing and frequency, and a solid knowledge of the player types around you are essential in order to both play well from this position, as well as play against others in it profitably.