I like many types of puzzle games, but at the moment, there are two particular styles that I'm looking for.
1) Avatar-centric environmental puzzles. These are the game where you control one or more characters and negotiate hazardous environments, typically a series of levels constructed from a set of consistent rules (switches, spike pits, statues that spit fireballs, etc). Some examples are The Lost Vikings, Adventures of Lolo, Deadly Rooms of Death, and Solomon's Key.
Note that I'm not presently looking for the "puzzle platform" games descended from Prince of Persia - Out of This World, Tomb Raider, Ico, Oddworld, Blackthorne, etc. I love those games as well, but they're a little more action-adventurey and a little less puzzly than what I'm looking for at the moment.
2) Abstract object-manipulation puzzles. This is not as specific as #1, but typically this sort of game gives you a playing field with a variety of items or objects to mess with and arrange in order to accomplish some objective or other. Lemmings, World of Goo, Obulis, and The Incredible Machine are good examples. Pipe Dream is a little too simplistic, and borders on the "parlor game" sort of puzzles found in e.g. The Seventh Guest (you want me to complete the knight's tour? Why?). Brickwell and color match games are not what I'm looking for either, nor is Minesweeper.
I'm looking for challenging puzzles, and I'm OK with twitch requirements. Of the examples I listed above, The Lost Vikings, DROD and Incredible Machine are my favorites (World of Goo was a nice game, yes, but it was way, way too easy).
Trine does come to mind as one that I have yet to play, but I can't see myself playing that one until the price comes down a ways. I only spend $20+ on games that I'm dying to play, and I hear Trine is not much of a challenge.