X-Com is one of the great PC strategy games that has never been truly replicated. It had a few spiritual successors, such as the Jagged Alliance series, but it's such a unique formula that it's hard to re-create in another format.
You are X-Com, the newly formed inter-governmental agency created to fight off a quickly escalating alien threat. Your basic goals in the game are to: Protect the world from alien infiltration (which in turn increases your funding from the nations you protect) and gain alien technology to research (providing you with better weapons and information about the alien threat - including how to destroy them). Everything else in the game is in support of these goals.
The game's scope is massive. You have base creation: A sim-city type minigame which can come into play should they aliens figure out where your base is. An admittedly weak minigame around intercepting UFOs. Your soldiers have RPG-type skills that improve with use. Research, engineering and purchasing decisions are often difficult due to money problems.
X-Com's main focus (and main draw) is the tactical turn-based combat system. Each soldier has "action points" (AP) which determine how much they can do in a turn. Shooting takes a percentage of your AP based on the kind of shot you're taking (An Aimed shot, giving the highest chance to hit takes 60% of a soldier's total AP for the turn). And movement costs a set 4AP per block traveled.
You might be thinking: "Turn based combat? That sounds archaic!" But moving your troops can draw the fire of waiting enemies (You can store AP that your soldiers will try to use to shoot at enemy movement as well) and the brief time between the shot being fired and the realization that it hit/missed your soldier is tension-filled. Similarly, clicking "End Turn" when you're almost positive you got the last alien can be harrowing.
Using cover in combat is essential and the environments are nearly 100% destructible. (Keep in mind, this game was made in 1993. How ridiculous is that?) Buildings, space-ships, fences and walls can all have holes blown in them if you use a large enough gun or grenade.
There are a few different types of missions as well, and the terror missions involving civilians can get downright nasty. The aliens will shoot and kill civilians and you have to try to keep as many alive as you can while wiping out the aliens.
The only complaints I have with the original game, are these:
1: The only way to open a door is to walk through it which opens your brave soldier up to being fired upon before being able to see his assailant (Fixed in later games).
2: The research stuff can get a little wonky (You have to capture specific alien leaders that are only available on certain ships/missions, and occasionally, you interrogate high-level prisoners out of order resulting in the need to capture another).
3: Psychic attack/defense is annoying as hell.
The game feels complete in a way that few games through the years have. Things tend to work the way you expect them to, and the world feels like a living, breathing place full of people. When you first reach a milestone - like successfully invading an alien base - you really feel accomplished and well rewarded with new technologies and equipment to research.