Civilization V is a complex game, featuring multiple aspects that you need to monitor to maintain and advance your civilization. This page attempts to tackle most of the game's concepts.
Cities are the core of your civilization. They serve as a point where you can produce units, construct buildings and create wonders. Cities are founded by Settlers, which you will usually start with at the beginning of a game, and its location will affect how useful it is. It is generally preferable to found cities near rivers and by the sea, as these location usually have plentiful Food for population growth. Once a city is founded, you can order it to build units, buildings and wonders, or simply purchase them using Gold. The first city you found will be your Capital, meaning that it has a higher output of Production, Science, Gold and Culture. Creating trade routes through Roads and Harbors to your Capital will also generate additional income. If your Capital is captured, one of your other cities will be made the Capital, until you retake the previous one.
There are citizens living in every city, which you can assign to work tiles in a three-tile radius inside its borders. Doing so lets you get extra Food, Gold and Production depending on the tile's yield. By default, citizens are assigned to work lands by a computer-controlled governor. The governor will generally focus on getting a balanced yield, although you can set it to focus on a certain aspect like Production for huge construction projects, or just manually assign citizens yourself. You may also assign some citizens to work as specialists, provided you've constructed a certain building. Specialists increases the output of one of your city's aspects depending on what type of specialist it is. For instance, a Scientist specialist helps the city generate more Science. Specialists also speed the creation of Great People, which will be covered in more detailed on that page. However, specialists working in buildings are not working tiles, so you will have a reduced yield from the tiles around you.
A city's borders will grow naturally at a rate dependent on the amount of Culture you generate, though you can manually purchase tiles yourself using Gold from the city screen. The borders of a major civilization cannot be passed by most units from other civs, doing so will automatically make you declare war on that civilization. You need to establish Open Border Treaties with another civilization to move pass through their borders peacefully. That means strategic placement of cities will let you control important chokepoints or to restrict the movement of other civs. It also forces civs to sign Open Border Treaties with you, which you can use to demand resources or Gold.
Cities can be captured during war. Each city has a health pool similar to a unit, which increases based on the number of citizens living in it, whether it is built on defensive terrain, whether there are garrisons in the city and the existence of defensive buildings like Walls. Cities are much tougher than most units, and attacking a well-defended one will usually result in a significant lost of health to the attacker. A city can also bombard units close to it. Once a city's health drops to zero, it can be captured by the attacker, but only by a melee unit. Even if you have surrounded a city with Compound Bowmen and Catapults, you cannot capture it without a Swordsman.
After capturing a city, you will plunder a short amount of Gold from it and get to decide what to do with it. You can annex the city into your empire, letting you control what it produces and letting you get the benefits from the buildings, wonders and tiles in the city. However, annexed cities generate extra unhappiness that can only be nullified by building happiness-increasing buildings or a Courthouse, which increases the maintenance cost of your empire and diverts production for a few turns. You may also puppet a city, which has no happiness penalty, but you won't have any control over what the city does. The city will only construct buildings, which can not only increase your maintenance costs unnecessarily, but also means you must provide units to the city to defend it. You can choose to annex a puppeted city anytime. Lastly, you may choose to raze the city, though this option is not available for a major civilization's capital. Doing so will completely wipe out everything in the city, leaving a pile of ruins behind that lasts for one turn. Razing cities will make you take a diplomatic hit from other civilizations.
Food and Growth
Food is a type of currency that is specific to each city, and it affects how fast its population increases. Every citizen in a city consume 2 Food every turn, and having a lower Food yield than the amount needed to keep citizens fed will result in starvation, which will cause your city's population to lower until the city can support itself. Food is mainly obtained by having citizens work tiles that provide them, although some buildings, Social Policies and religious beliefs also grant additional Food.
Every turn, all the excess Food will be added to what is called the "City Growth Bucket". Once a certain amount of excess Food has been added to the bucket, a new citizen is born, ready to be assigned to work a tile or become a specialist. However, any leftover excess Food in the bucket will be dumped, meaning that you must start from scratch to accumulate Food, though the loss of food can be reduced with certain buildings. You can check when that your city will likely grow by seeing the small green bar on a city's banner, or the Growth meter in the city screen. Note that there are some other factors that influence Growth. Producing a Settler will cause halt the process completely, while having a mildly unhappy city causes the city to grow slower. On the other hand, having a city hold a "We Love the King Day" by connecting a requested resource to your trade network will increase its excess Food by 25%.
Happiness is a measure of how happy your citizens are, and keeping them happy is important for running a successful empire. Instead of being measured on a per-city basis like previous games, Happiness is now a global value. Happiness can be raised by discovering natural wonders, improving or trading luxury resources (though note that you will only get Happiness from one copy of a resource), certain buildings, wonders, social policies and religious beliefs. You also start with a certain amount of Happiness in the beginning of the game depending on the difficulty level. Unhappiness obviously lowers your Happiness, and serves to limit you from growing your empire too freely. Unhappiness increases with the number of cities you have, your population in each city and ehther you have any annexed cities.
Keeping your citizens happy has its benefits, as excess Happiness will be added to your Golden Age counter, making a very happy empire trigger Golden Ages more often. Conversely, if your citizens are unhappy, the empire suffers. A mildly unhappy empire represented with a frowning face on the happiness counter will have significantly reduced Growth. In an extremely unhappy empire, represented with an angry face, Growth halts, you're unable to produce Settlers and all of your units get a massive combat penalty.