I'm surprised to see so much love for slider puzzles, I thought those were the most common hate objects of this community. I usually get angry when a game presents one to me, but really, the ordinary 3x3 slider is fairly quick to get past after so many games.
I had some difficulty pinning down the type of puzzle I enjoy the most. I suppose you could call them "communication puzzles", or something. It's when you need to communicate with the game or with a character, but with less restrictions than in dialogue trees. I'll give some examples to clarify:
- Gabriel Knight 1, where you construct a voodoo code message
- Journeyman Project 2, at the end where you need to guess someone's password
- Programming the droid at the beginning of The Dig
I like decoding/deciphering, but when you also need to encode something with the code you've learned, there's an added sense of accomplishment. Especially if you need to use it against an enemy, like in GK1 or JP2. I wish I could explain it better.
Another example could be the search engine puzzle in Still Life, if it hadn't been so poorly hinted at. Also, I once played a free game on-line about a space traveller getting stranded in an alien world. In order to get off the planet, you had to listen in on the inhabitants and try to decipher their language, and finally be able to communicate with them. That was pretty cool, almost completely made up of my favourite kind of puzzle.
