The story continues immediately from where Alien left off - Sigourney Weaver is back and is roped into removing the 'Alien' nest following a tragedy involving a human settlement being used to host larger numbers of Aliens.
Firstly, it should be noted that Aliens is a completely different genre of film to Alien. There are still elements of the original's anti-climaxes and tension-building but the action sequences are much more explosive and a lot less subtle. More Aliens generally equals more bullets.
Unlike most sequels, this fits into the storyline brilliantly with the previous film, expands on the theories set out in the original but doesn't add or remove anything too over the top.
A shining example of how a sequel should be made.