Hat collecting is the process of gaining rights on Wikipedia (and other wikis) just to show off or to boost one's ego, rather than because you have any actual use for them. Voters in the request for adminship process need to decide whether or not the user requesting adminship actually has a use for those rights or is simply "hat collecting". This is why adminship candidates should have some clear idea why they want to become admins, rather than simply the feeling that this is what one does at a certain stage in one's Wikipedia career.
Not having a particular user right doesn't mean you are somehow deficient. Lots of people don't have file mover, but that's because there's only a small community of people who care about maintaining the file namespace. But on the converse, having file mover doesn't mean you are an expert on file moving, it just means you've been granted the ability to do it. For some administrators, handing out rights to a reasonably trustworthy user is a very simple way to prevent them from persistently making requests of that form.
Of course, one should be very careful before alleging that someone is hat collecting. Simply responding to everyone wanting rollback or even everyone wanting to become an administrator with the accusation that they are a hat collector marks a failure to assume good faith.
If you are applying for rights, be sure to show why you think you can improve Wikipedia by having those rights, generally by showing a positive record of actions and attitude. Don't just assert that you'll find the tool useful, give evidence of why you need the tool and you'll dramatically reduce accusations that you are simply collecting hats.
Don't just collect hats, wear them after you get them!
Hat collecting can occur across wikis, with users using the rights they've got at one wiki to bootstrap an application on another wiki. This isn't always a problem: most of the user rights require trust. If a long-standing Wikisource admin turns up, there is a good reason to presume trustworthiness compared with a completely new user.
But remember, some wikis are, or were, very lax about handing out user rights. The Simple English Wikipedia once gave a user rollback after two edits... on the basis that said user also had rollback on Commons. (Admittedly, this is no longer done.) Likewise, some rights like autopatrolled or reviewer have many different technical meanings on various wikis, with vastly different associated levels of trust (some higher than here, some lower). Some even automatically allocate extra rights, like the English Wikibooks, which gives out rollback and reviewer by an automatic process. Be careful not to place too much weight on rights on other projects when considering granting them at the English Wikipedia.