I really enjoy strategy games, but only of the "turn-based" variety. I believe that turn-based strategy games get the heart of the genre better than RTS (real-time strategy) games do. After all, strategy is about thinking and planning, it's not a race.
Think about the greatest strategy game of all time: chess. It's turn-based. You have to watch you opponents moves and react (and even predict) appropriately. A pure battle of the minds.
Maybe I'm just a little biter because I'm not terribly good (in fact, downright terrible) at RTS games (Warcraft, Starcraft, Command and Conquer). I find that when I'm playing these games, especially when playing against human opponents, I am never able to think. With everything continuously and constantly happening, I don't get a chance to properly plan ahead.
Now before you cry out: "But that's the whole point of RTS's!!!", let me just say that I understand that, I'm simply pointing out how it's not for me. I'm simply saying that to me, a pure strategy game comes down to thinking and completely removes elements that rely on reaction time and precision. If I lose at a strategy game, I want it to be because of my poor decisions or an oversight (which is completely my own fault), not because I mis-clicked on a unit on screen and didn't give him orders fast enough, or worse, because the other guy has created a key-bind for every unit type in the game and I haven't.
This recent revelation of my love of turn-based strategy games has come after I finished the campaign in Advance Wars: Dual Strike for the third time. I also came to the revelation that it is hands down, my favourite DS game.
The Wars series (sometimes referred to as the Nintendo Wars series) is a series of turn-based strategy games that have long existed in Japan but were never released in North America until Advance Wars on the Game Boy Advance (previous entries were released in Japan for the Famicom and the Game Boy).
While the games have a "cartoony" look, they are very intricate and expertly balanced military strategy games. I definitely count the Advance Wars games as my personal favourite turn-based strategy games.
From a game design point of view, my appreciation of strategy games comes from the fact that the outcome is almost always decided by the players understanding of the game (or lack thereof). This is different from action games, where perhaps the player has lost because of a slip of the thumb, causing them to miss a jump or fail an attack. The reflex element (some would call it skill) has been completely removed, leaving the player with only their wits to help them get by.
That's why I (heart) strategy games.