Addictive, satisfying TBS-fun.
I know there is confusion, and many people compare it to AoE, but that's about the stupidest thing one can do.
Why? Read on:
1) This is a TBS-game. Everything happens in turns. You build your cities, expand your territory by achieving cultural goals, you build your army to try and crush your enemies (or at least you try to defend it against people that try to crush you), ... But this happens in turns: 2 turns to construct a Warrior unit, 15 turns to build a building, 7 turns to research a technology,... You get the picture.
2) Different ways to achieve victory. You have about six victory conditions that can be met (you do not have to accomplish six goals at once), instead of just 'wipe out your enemies'. You can accomplish victory by having the highest score at a certain point in the game, you can accomplish victory by wiping out all your enemies, by having 70% of total world population + 40% territory, get elected by the UN as world leader, by having three Legendary cities (cultural), or be the first to send a colony into space.
3) From 4000 BC to 2050 AD. In the first Age of Empires, you started in the prehistory and could play until the Iron Age. In the subsequent games, you had different ages that you could achieve, but you could never go into the Space Age for instance. In Civ4, this is possible. It is possible to fight off the US Navy Seals with Longbowmen, although you're never sure that this will work.
4) Diplomacy is essential. Of course, you can declare war upon every single civilization in the game, but having them all at once would mean your utter demise in less than 30 minutes. You have to carefully select your allies, decide whom you're going to approve to enter your lands and so on. In Age of Empires, you could very well manage alone, but in Civ4, it's more or less impossible.
5) It's addictive. I've always found that Age of Empires lacked something really interesting to boost the addiction factor, but in Civ4, you have the well known 'one-more-turn-and-I'm-off-to-bed' - feeling, and you will still find yourself playing at 6 o' clock in the morning...
Of course, it's not all roses here: it's definitely not for everyone, and the graphics are a bit underrated nowadays. Also, if you're looking for a historically correct game, you'll have to find something else to play, as it's very well possible that George Washington will be born around 2000 AD in Arabia, which was (of course) not the case.
Oh, and one more word about the demo: the demo does not give a good overall view of the game. Try to play it at a friend's house or buy it (you can get it for a penny nowadays).
Conclusion: if you're into TBS-games, citybuilding, management and history, you should definitely get this game. If you expect an Age of Empires-clone: buy Age of Empires III or Empire Earth...