Besides Food, Production and Gold, your Cities can collect something else: Culture. Culture in Civilization IV is generated somewhat differently than in the game's predecessor: the Culture tax slider and the number of Artists in a City determine the base amount of Culture that it generates. Some Wonders add to it as well, but some other Buildings, Wonders (e.g. National Epic) and Civics (e.g. Nationhood) act as multipliers for this base value. Great People can also boost the culture value of a City. Culture has three main effects on gameplay. The most straightforward is that it helps you to accomplish a Cultural Victory: if you manage to get three Cities over 100,000 Culture, you win the game.
Another advantage that comes with Culture is Borders. Without Culture, the Borders of your empire extend only one tile around each City. But when the Culture value of a City reaches a certain threshold, the Borders expand. When they reach the next threshold, they expand again. Rinse and repeat, eventually connecting all your Cities into one large empire. This has a number of advantages: it allows your Units to move freely and it blocks all enemy Units (including Settlers and Missionaries) from moving near your Cities (unless they declare war or sign an Open Borders treaty) or at least slows them down (when at war). When used near chokepoints, this can be used to cut opponents off from entire subcontinents. Resources also need to fall inside your borders in order for you to be able to claim them, as do the tiles that a City can work.
When the Borders of two rival Cities clash, the City with the strongest Culture wins and when strong enough can eventually 'beat back' the opponent's Borders and even completely enveloped the culturally weaker City [see image] -- something that was not possible in Civ3 where you were always guaranteed a certain minimum amount of territory around your Cities. In Civilization IV it's even possible to completely surround a City with another Civ's Borders, creating a San Marino or Monaco-like enclave. City flipping from Civ3 still exists but has been toned down: as Borders expand close to rival Cities, the Citizens in that City will become more and more unhappy -- they want to join the more cultured Civilization. Eventually the City will revolt for a few turns. If this happens twice, a City will split off from the original owner and join the culturally dominant Civilization. This means that players have ample warning and time to act to prevent these flips from happening. But often it doesn't even come this far, as these Cities become so inefficient to maintain and such a burden for the owner that (s)he would be better off giving it away through diplomacy. Combined with Great Artists, which can give a large one-time Culture bonus (4,000 points), this can be a potent strategy and a viable way of 'peaceful' conquest (though of course it could anger a player into declaring war).
The third advantage of Culture is that it strengthens a City's defenses. In Civ3 bigger Cities got bigger defense bonuses, in Civ4 this is no longer the case: now more cultural Cities gets a bigger bonus instead. This rewards players who focus on cultural development rather than solely on growth. It also makes it more difficult to defend recently conquered Cities: once a City is conquered, the Culture value is reset to zero and any cultural defense bonus it might have had is gone. This makes it much more vulnerable to (re)capture, so players will have to invest more if they want to keep their conquests.