Cities are still the cornerstones of your Civilization. The City is where you people live, where you collect resources and where you can order the construction of Buildings, Wonders and Units. Cities occupy one tile on the map and can collect Food (for growth), Hammers (formerly Shields; for production) and Gold (for unit support, research, culture, diplomacy, etc) by having its Citizens work the tiles surrounding it. Which tiles are being worked can be seen on the main map: if they have Improvements, these are animated, if not, a little hut or ship will be visible on the tile.
As in Civ3, you can build all sorts of different Buildings and Wonders in your Cities for a variety of different bonuses. All these Buildings and Wonders will be visible on the main map. Examples of this can be seen below: in the bottom left of the City in the first image, the Pyramids Wonder can be seen and in the center an Obelisk Building is visible. In the last image Walls are visible. There are a total of 102 Buildings in the game, among which 40 Wonders: 28 World Wonders (which can be built only once) and 12 National Wonders (which can be built once by every Civ). Civilization IV: Warlords will add 3 more Wonders and a Unique Building for every Civilization. No details have been announced yet. See the Civilopedia for a detailed breakdown of all the Buildings and Wonders in the game.
Other information about a City that can be seen from the main map is it's name, population, whether it's a capital (star around population number) or not (circle), what it's building (both in graphic and text form), what Religions it has and what kind of defensive bonus [see images].
In your Cities live your people, in the form of Citizens. To make sure your people keep working at maximum efficiency, you have to keep them both happy and healthy.
There are many factors which affect Happiness: the Buildings in a City, the Wonders you control, what Religions your people have (if any), your Civics and those of other people, how many Luxury Resources you have, how what portion of income is dedicated to the Culture tax slider, the number of Future Technologies you have, etc. If Citizens are unhappy they won't cause riots once there are enough of them, as was the case in previous incarnations of Civ, but they'll just stop working right away. The happy Citizens remain unaffected, but the unhappy ones will not contribute any resources to your City.
Health is a new concept in Civilization IV and is intended to replace pollution from Civ3: it's a much more comprehensive system and works on a City level rather than a global one. It works pretty much along the same lines as Happiness: it's a separate value for each City and has a large variety of positive and negative influences. Terrain is one of these influences: proximity to Fresh Water or Forest adds a Health bonus, as does every Food Resource (from terrain or trade). Future Technology gives a Health bonus as well. The amount of Food that is available in a City also affects Health. So do Buildings: industrial Buildings like Factories decrease Health, while others increase it. If your Health value is negative, you will lose Food income, which limits the speed at which your City can grow, or it can even cause your City to shrink if the problem is big enough or you don't have much Food income to begin with. As such, the Health value is the primary factor in determining how large your City can become.
Every city can also have a number of Trade Routes. At the start of the game there are none, but as your empire expands and you invent new technology, adopt new Civics, build new Buildings and Wonders and and meet new Civilizations, trade routes with other cities will automatically be created, the amount of income of which depends on the distance to your neighbors, size of both cities involved, technology, Buildings available, etc.
Building maintenance cost from previous Civ games is no longer present in Civilization IV: you can build as many Buildings as you want, they don't cost anything. Instead, you now pay Maintenance for every City you own -- and with every new City you build or conquer, the Maintenance cost of all your Cities will increase. This makes ICS or REX tactics from earlier incarnations of Civilization unfeasible in Civilization IV. You will have to develop your Cities before you can build new ones, or else you will soon go bankrupt.
As in previous versions of the game, your can convert your Citizens to Specialists in Civilization IV. However, the types of Specialists available to you have changed. The following Specialist types exist in Civ4: Citizen (gives 1 Hammer), Artist (gives Culture), Scientist (gives Beakers), Engineer (gives Hammers), Merchant (gives Gold) and Priest (spreads Religion). The exact amount of resources these specialists can yield depends on what Civics you use and what Wonders and Buildings you have. All these specialists except Citizens also contribute Great People Points [see Great People]. Note that you can't assign as many Specialists as you like, you need to build certain Buildings or Wonders or adopt certain Civics to be allowed to place certain Specialists. For example, a Temple allows you to turn 1 Citizens into a Priest.
All in all, City management has changed quite a bit from Civ3: pollution, corruption, maintenance and rioting are all no longer present in Civ4. As mentioned, pollution has been replaced by Health. Rioting has been reduced from a City-wide event to individual Citizen level. Building maintenance is gone, instead there is a per-City Maintenance cost, which increases as you build more cities, a system which renders corruption unnecessary. The Great People system still makes specialization important (as opposed to building everything everywhere), the original function of Building maintenance in earlier Civ games.
For new players and rulers of enormous empires there is not need to worry that things will be too complex or require too much attention: a lot of attention has gone to improving the City Governors, so leaving the management of Cities in the hands of the AI should be a viable option.