Many movies have different versions of them. Most typically there's the so-called "theatrical cut" (or "theatrical release"), and then some kind of "director's cut", or "extended cut", which is released later, usually on home video (and usually as an alternative to the original theatrical cut).
A director's cut most typically (and in many cases most lazily) simply adds some scenes that were originally filmed but not included in the theatrical release version. These may include longer, extended scenes, or completely unseen scenes.
More involved director's cuts may go far beyond that, replacing some scenes with alternative takes, re-ordering scenes, and sometimes even adding material to the movie created after the original release (such as new computer graphics or other effects).
People often tend to think that the "director's cut" must be better than the original theatrical cut. The notion is that the original cut has often been done under pressure, both in terms of time and at the hands of meddling studio executives and producers who love to chime in and force the director to make the movie like they want, rather than what the director wants. The director's cut, however, is the product of the director being allowed to do whatever he wants, without pressure and without studio meddling; to realize what his original vision for the movie was. The director is allowed more time to work on the movie, without pressure, and do more freely what he wants. Therefore the end result ought to be better.
Or does it?
Sometimes that's indeed the case. Sometimes studio executives have outright gone against the director's will and quite radically changed the movie in some way, such as changing the ending, because the executives didn't like the original ending. The director's cut then restores this original ending that he (and the scriptwriter) intended. In some/many cases this director's cut is lauded as being better than the theatrical version because of this. (Blade Runner might be one of the most notorious examples.)
Othertimes it's not so much that the hand of the director was forced for the initial release, and only afterwards let loose, but it's simply that afterwards the director is not so much under time pressure and can now put more time on a better cut of the film. To fix things that ended up being a bit unclear, missing, or misleading in the original cut, to change takes with alternative ones that better reflect what the film tries to convey, to add extra scenes that make things clearer, to make the flow of the narrative better, to expand on the story, and so on. I think that The Lord of the Rings trilogy is a very good example of all of this.
(As one particular example from the first movie, the theatrical version makes it look like Gandalf knew that Bilbo was going to use the ring at his birthday party, and laughed when he did so. Peter Jackson later thought that this was actually a bit illogical and out-of-character, and didn't really make much sense, so in the Director's cut he re-edited that scene in such a manner that Bilbo using the ring came as a surprise even to Gandalf, who got visibly upset about it. It indeed works better this way and makes more sense. The ring was not something to toy with.)
But are director's/extended cuts always better than the original theatrical cut? Not necessarily. Sometimes the director's cut can be arguably worse than the original.
Sometimes this can be because the "extended cut" is nothing but a lazy attempt at cashing on the film's success by releasing a different version of it, lazily inserting unused scenes here and there, with little regard to whether they really fit there or not. In the absolutely worst examples I have seen, the extra scenes actually have a different visual/audio quality to them compared to the rest, making them stick out like a sore thumb.
For example, the extended cut of Blues Brothers is arguably worse than the original. The extra scenes are unnecessary, superfluous, and in fact some of them introduce continuity errors (such as the clothing of some characters changing between scenes without explanation.)
The Director's Cut of Aliens is a mixed bag. Some additional scenes are generally considered good additions, while others are considered by many to be superfluous, and in fact detrimental to the overall narrative flow. The most prominent examples of the latter are the early scenes shown of the colony, before the marines arrive (which were not in the theatrical release). Not only are these scenes superfluous and detrimental to the narrative flow, they in fact pretty much spoil the nature of the emergency. (Not that there's much to spoil in the meta sense, but if we look at the movie on its own right, ignoring what we know outside of it, these scenes act pretty much as spoilers.) The film is supposed to be very suspenseful up until the marines first time encounter the aliens, but this extra scene spoils that suspense pretty much completely.
Personally I much prefer the original cut of the movie over the Director's cut.
Director's cuts are not always better than the original, even though we have often conditioned ourselves to believe otherwise.