Can you actually have different types of playthroughs depending on what kind of "base" you build (in other words: are there real choices to be made?), or is it more like just some random stuff you have to micro around for some quasi-RPG?
If there is replayability in this, and it is SP instead of MP, and i can play it on my lame machine, then it sounds good.
It does actually matter how you build your bases in most of the games, the original X-Com and most of it's later clones. If you want to play a some-what modern game that's as close to the original X-Com as possible I would go one of two ways until Xenonauts is released (will talk more about this in a second). As far as bases go though basically you have a limited amount of funds. Building installations like a "Alien Containment" so you can trap live aliens, or building more hangers for interceptor craft, anything really, all costs money. In some of the games the bases you get may be limited in size and you might not be able to build everything you need in that one location so strategic placement is important. It depends a little though on if you're playing X-Com or one of it's later clones.
To be as faithful to the original as possible, and I'm also sure both of these will run on your system, try either:
UFO: Extraterrestrials (make SURE to use the BMan mod as it fixes a lot of things the devs didn't)
UFO: Aftermath (the first game in the UFO:After series)
Both of those do a pretty good job of sticking to the feel of the game-play of the original X-Com and both of them manage to not look too terrible in the process. Don't expect mind blowing 3D or advanced lighting effects though. Think more like Fallout 2 or Baldur's Gate type graphics only with slightly newer graphics.
To add to what Shadowtongue has said, there is indeed a LOT of replayability in most of these games as it's how you play the game that makes the story. They aren't heavy with RPG elements, but they do have soldier progression because they have stats they gain at each level and rank and you train them. Some of the games have training specializations soldiers can have once they meet the stat requirements, like "Sniper" or "Medic" (the "UFO:After series does this).
There are certain points when you are researching the aliens or their technology where you make breakthroughs in what would advance the "Story". In many of these games that might be when you take a alien of high enough rank alive in a mission and interrogate him when you get him back to your base. Another might come when you've finally found a certain weapon the aliens have been attacking you with and when you research it you unlock a whole new branch of technology that lets you make new armor, weapons, and even better vehicle parts for your troopship and interceptors. Again it's how you play the game that makes the experience or story.
As for another new game that's coming that is remaining truly faithful to the original X-Com experience, I HIGHLY recommend "Xenonauts". It should be out this year. It's got a great setting and time-line of aliens invading in the 80's. The feel of the game is simply amazing really and it has a great atmosphere. The game-play features cover all the areas where the original X-Com shined and they're adding some newer updates that stay very true to the genre experience. You can find Xenonauts on Desura as well and get in on the Alpha too. Also it's very affordable and will run on just about any system. Again don't expect mind blowing graphics as Goldhawk Interactive is a very small Indy company. Basically one guy, a English dude named Chris, put this all together and is the whole company (plus the people he's hired to do the work of course). I know he's dedicated to his company though because he's sunk his entire life savings and inheritance into making Xenonauts. The game doesn't look horrible graphically, mind you, just don't expect anything too fancy. You can see footage from the latest build on YouTube.