A type of wiki contributor, a wiki curator acts in many of the same ways a museum curator does. Maintaining the artifacts (the articles), researching them, arranging them into interesting dioramas (the indexes), and so on. And you aren't allowed to touch any of them.
There are generalist curators and specialist curators. Specialists have a deep interest in a specific medium or a specific genre and don't do much curating outside that interest. The interests of generalist curators are wider. Or perhaps they just wander.
If a wiki isn't attracting curators of either type, it just isn't working.
How a curator works on TV Tropes is by tracking certain pages on their watchlist and looking out for things like bad examples and clumsy edits. The ultimate goal of the curator is to protect a page from eventual action in the Trope Repair Shop.
Curator is an entirely informal term. To become one, all you have to do is pick a page you like and check its page history every so often to make sure that Wiki Magic is making it better. Because it's one of the more positive things an editor can do, you should take some credit on your contributor page for the pages you watch out for.
Since this is subject to Sturgeon's Law, you should also watch out for yourself, and how you approach this role. Be sure you're wearing the Curator hat in the wiki's interest rather than your own, and know when to exercise personal restraint in editing.