The director's cut in Aliens is much akin to an 'extended' edition. Extra scenes added, and it increases the tension in what is otherwise a much more action oriented film. The theatrical cut is totally fine with this film, but the new cut is worth watching too.
The one in Alien 3 is called the Assembly Cut, and it's an attempt to make the film more in line with director Fincher's vision of the film, after he received a lot of studio interference. There are alternate plot lines, and sub-plots that were cut have been restored. It isn't a 'Director's Cut' exactly, as Fincher has disowned the film, but it improves a deeply flawed movie in my eyes.
As for Alien Resurrection, if you decide to watch it, there is very little difference between cuts. There's an alternate opening in the director's cut, complete with a horrible CGI space insect.
