A brief description.
You start out with 2 units, a settler and a worker, on a totally unexplored world. Using these two units, you build a city, find resources, build military units, build city improvements, build more cities, meet your neighbors and trade and war with them, research new technologies, and attempt to become the best civilization in the world.
Each civilization that you play starts with certain advantages. Some can build libraries more cheaply so they can research faster. Some can build temples more cheaply to keep their people happy. Some have workers that can build roads and mines more quickly.
You must constantly balance your resources to keep ahead of your computer opponents. If you don't spend enough on science, you may find yourself defending against tanks with spearmen. You must find new resources in order to build new units. If you don't have oil, you can't build tanks, so you either have to find it, or trade for it.
You must decide which form of govt you want to use. Democracy gives you lots of cash, but can be difficult to maintain, because the people are easily made unhappy during wars. Communism is great for wars, but lousy for researching new technologies.
You can play the game right out of the box, and use the tutorial to learn the bare basics of the game, however, for the first 5-10 games, you will be learning new aspects of the game, things you had no clue about in your previous games

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I've played Civ I through Civ III, and Civ III is the best version yet. The best games to compare it to are probably Age of Empires (which is real-time, as opposed to turn-based Civ III), and Sim City (cause you are always building and improving your cities).