X-Com is one of my all time favorite strategy games, excellent atmosphere, interesting strategy, challenging and fun
The main things that I really like about X-com are:
Atmosphere/environment: This game just feels cool... dark and suspenseful, a little campy, gritty, desperate..
Turn based combat: With line of site, destroyable environment, great weapons, many tactical options, combat is hard and fun. I like that I had to learn tactical strategy to win (cross fire, grenades to drive enemy out of cover, smoke grenades for cover, etc...)
Strategy (with role playing elements): The strategy part of the game... including the world view, choosing base locations, building bases, research, soldier advancement, etc... all very satisfying
Of course the graphics are not good compared to anything current and there are elements of the game that are a bit over the top in difficulty (some early missions before you have proper equipment and numbers can be very hard to survive let alone keep most of your team alive).
Still waiting for a game to actually capture all of the excellent elements of this game and building on it with current technology.