This was one of the few games I played over and over as a teen.
I'll be honest, I never got into the sequel, because I didn't like the way the planets were depicted (though it probably made influence more intuitive). So I don't know how this one holds up to GalCiv II. But on its own merits, it's a good game.
In Galactic Civilizations, you start out as ruler of united Earth. The various alien races, having gotten the secret of hyperdrive from the humans, have turned off their massive wormhole portals and scrapped them for hyper-capable colony ships. The real space race has begun, and it's up to you to determine the path you want humanity to take.
Madcap conquest is definitely an option, but if that either doesn't sound appealing or isn't practical, you can turn to other means - being a technocrat and staying ahead by selling and leaking techs; buddying up to the top dog; making everyone's economies dependant on your trade; or amass cultural influence by convincing the aliens how awesome Hollywood, Nike and Coca-Cola are. In practice, most games tend to involve a combination of all the above - the alien races tend to be more receptive to your diplomacy if you have a lot of ships with big guns, or if a significant chunk of their economy is powered by trade with you. And being ahead in tech is useful for keeping a welfare economy afloat by selling techs to other empires, or to prop up an enemy of an enemy without going to war directly. Conversely, rolling in dough means being able to bribe your rivals, buy state-of-the-art warships from a militarist friend, and ramp up domestic spending (including research and social projects).
Along the way, you get to decide whether to be an evil empire, a benevolent federation or any combination and variation of them. A sprawling but sparsely populated expansionist nation, or a handful of highly-developed star systems? Military juggernaut or trade linchpin of the galaxy? Underhand arms dealer or open warmonger? Your answer to these questions is often simply "yes". It's telling that a game that allows you to build aphrodisiacs and massive theme parks also give you the option of building propaganda machines and pseudo-Death Stars - often at the same time (who says star-killing not-moons can't have fun water slides?).
The alien civilizations all have distinct personalities - by default some are good, some are neutral, some are evil, but fiddle with their alignment setting at the start of the game and they change accordingly - a civilization that originally extended a friendly hand will now threaten to exterminate you at the slightest excuse, and vice versa. They will gloat when they're winning a war, grovel when they're beaten; extort you when they're much more powerful and don't like you, give you periodic gifts with mincing courtesy when you're the powerhouse.
Though a thoroughly old game by now, this game has plenty of charm and solid mechanics underpinning them. I'm sure nostalgia only affects my verdict -that is still amazingly fun- only a little.