I haven't played every 4x game out there, but here's my take:
Master of Orion
The original and still the best. It's got just the right amount of micromanagement. You run your planets via a simple slider scheme rather than building specific planetary improvements which keeps huge empires from slowing down too much. The ship design aspect is also a lot of fun. I also think that the graphics hold up pretty well for an old DOS game. On the downside, you will need the manual to get past the copy protection.
Master of Orion II
Some people love this but I don't think it's very good. The problem is that instead of taking the MoO approach and striking a balance between the micro- and the macro-game, the developers (Simtex, IIRC) decided to throw in everything plus the kitchen sink. So instead of just setting a colony's industry/economy/ecology you have to actually build individual improvements, and instead of a single colony per star you now have to deal with the actual planets in the solar system. One kind of neat touch is that you can hire heroes to run your colonies and fleets, but overall I find the game doesn't have much character - it just becomes a race to build the next +25 colony improvement or whatever.
(new) Galactic Civilizations
This game is pretty much a copy of MoO2 but with a few improvements. It's got all the same flaws, but there are a couple of good ideas in the game. For example, there are four civilization types so if you are running anything other than an Imperium your government can get voted out of office (you lose a few bonuses if this happens). There are also trade routes, something I'd never seen done in a space 4x game before. Basically you have to scout out wealthy planets from other civilizations and send your freighters to them. This adds a whole other dimension to the game because trade routes are quite lucrative so when war is inevitably declared you have to actually protect your territory.
However there are a bunch of other problems: the tech tree is broken, the United Planets resolutions have no effect (JUST LIKE REAL LIFE :downswords::downswords:), the ai is way too aggressive, there's no ship design minigame, culture victories are too easy while conquest victories are too hard. But that's okay because why would you want to play GalCiv when you could play...
Galactic Civilizations II
This game is the bomb. It's as good as Master of Orion. They gave it a nice 3D interface, so you can zoom all the way out and view the whole galaxy or zoom all the way in and see your spaceships in orbit. Stardock fixed basically everything that was wrong with the previous version, and if you're not happy with the base game there are a ton of options you can use to configure the gameplay, and if you're not happy with that the game is easily moddable (from a gameplay perspective).
The only real issue I have with GalCiv II is that the ai is still way too aggressive. Basically as soon as the game starts you need to start preparing for war - you're either stronger than the other races and they'll do whatever you want or you're weaker and they'll be demanding tribute ever other turn. Also, the culture victory is still too easy so influence races have it easy whereas war-faring races are gimped. Still a great game, though.
Alpha Centauri
I don't feel like writing a whole lot about this, but Alpha Centuari does have one of the best endgame payoffs ever in that you can build up an arsenal of ICBMs and nuke the world.
COUCHMAN IS BACK BABY