This article will give you information and advice on settling a city in Civilization V.
Basics
In order to settle (or found) a new city, you need a special civilian unit called a Settler. This is a standard land unit that is available right from the start of the game, but can only be produced in cities with a minimum 
Population Population of 2. The Settler is used by moving him to the location where you plan to settle, then pressing the "Found City" button; this consumes the unit and adds a new city to the map on the same tile that the Settler previously occupied. You may assign 
Production Production projects to your new city immediately.
You can found as many cities as you want to (or have space to), but there are a couple of considerations to be taken into account, which will be explained below.
Cities have to be at least 4 tiles apart from each other. That includes your cities as well as other civilizations' cities, and it prevents players from littering the map with cities. Other than that, cities may be founded on any tile to which a Settler unit can move (including a 1-tile island).
Lastly, you can't settle a city on a tile containing City Ruins (where a city used to exist until it was razed).
Message shown when trying to found a city using Settlers.
Technical Details When Founding a City
When you found a city in a particular tile, this tile becomes the basis of its production capabilities. The founding process adds some 
Food Food and/or Production Production to the tile's potential, so that each city has at least 2 Food Food and 1 Production Production output. So, for example, if you settle a city in a desert (0 potential of any kind), the city tile will start producing 2 Food Food and 1 Production Production (more observations needed). This is so that the new city has at least the means to feed a minimum population and produce something. Of course, if there is more considerable potential on the tile, this can only help the new city.
You can found cities on top of resources. This doesn't mean you'll be unable to use them - on the contrary, you immediately gain access to the resource when doing so, as if you've just built an improvement on it. Note, however, that you won't gain access to the resource if you haven't researched the technology which allows your empire to access it! For example, if you settle a city on a tile with Wine, you won't have access to it until you research Calendar.
Furthermore, you'll gain access to any bonus buildings tied to this resource, and to any benefits directly affecting the resource. For example, if you settle a city on a tile with Iron, you can then build a Forge in the city, and building it will increase Production Production on the city tile by 1. However, you won't gain any benefits that are related to a particular improvement for this resource - for example, you won't gain an extra Production Production point after researching Chemistry (which adds +1 Production Production to all Mines).
Every city always works its main city tile. New cities start with 1 Population Population, which means they can work 1 additional tile besides the main city tile. The greater a city's population, the more of the surrounding tiles it will be able to work.
New cities enlarge your empire's territory, adding all surrounding tiles to it. Note that this initial territory won't start expanding until you produce at least 1 
Culture Culture in the city.
Empire Consequences When Founding a City
Founding a city (as well as conquering one and adding it officially to the empire) has some negative consequences for your empire that you need to pay attention to. These are as follows:
Culture Culture points needed to adopt new Social Policies increase. The exact calculation is complex, and involves some bonuses, but the amount is somewhere between 5 and 10 percent of your current CP total for adopting the next policy. Note that the effect is immediate upon founding the city, so you may wish to delay settling the next city at least until you adopt the next policy.- Science Science points needed to discover new technologies are increased by 5% per city after the first.
- Unhappiness Unhappiness in the empire increases by 3 per city. Also, don't forget that every Population Citizen adds 1 Unhappiness Unhappiness - this is particularly important when conquering cities!
- The next Golden Age Golden Age counter also increases. The more Golden Ages you've had already, the more the counter increases per city. Before the first Golden Age, the increase is something like +5 per city (confirmation needed).
- Great People counters increase (no longer in effect with the expansions).
Where to Found a City
Choosing the right place to found a city in Civilization V is very important, and determines to a large degree how successful your empire will be. You should think about the following things:
Resources
To have access to any resources, especially strategic and luxury ones, you need them to be within your territory. Founding cities near them (1 to 3 tiles away) is the best way to assure that. Never forget that a city may only work tiles that are at most 3 tiles away from its main city tile. So, when choosing the spot, try to include as many resources within 3 tiles of it as you can. Even though your budding city won't include them into its territory yet, it will start growing it later with exactly this in mind.
Many times a resource may cause territorial contest. When there are two cities (one yours, one belonging to another civilization) in range of a resource, you have to be wary - the other civilization will try to take over the resource. In this case it's best to spend some 
Gold Gold to purchase the tile in question. This will cause some diplomatic outrage, but it's better than to lose the resource.
Note that strategic and luxury resources can actually be accessed WITHOUT being worked by a city - you just need them to be inside your territory, and build the relevant improvement on them. However, there's a catch: the automatic territorial expansion, based on Culture Culture, tends to expand your territory mostly 3 tiles away from the city. It WILL expand further, but only AFTER it has encompassed all tiles within 3 tiles of the city (though there are some extremely rare exceptions to this rule). Also, you can't use Gold Gold to purchase a tile which is more than 3 tiles away, which means that if you want to make sure you'll have access to a resource, you had better follow the rules described above.
Natural wonders are also a type of resource as far as founding cities is concerned, so follow the rules above to claim them.
Sea Access
Any civilization needs access to the ocean - this ensures it will be able to build a fleet and explore the world easily, and also conduct sea trade. Additionally, there are plenty of resources to be found on coasts. Explore coasts and find a suitable place for a coastal city ASAP. For a city to have access to the sea/ocean, it has to be founded on a tile bordering the water.
Note that not all large bodies of water connect to the world ocean! Some of them are so-called "inland seas," which are surrounded on every side by land. Settling cities on their coasts won't do you much good, unless there are important resources to be harvested nearby.
Some civilizations are marked as "seafaring." They will start the game already next to the sea, and most of the time their 
Capital Capital will be on the sea. Those civilizations will benefit from settling cities inland at first.
Fertile and Productive Land
The next thing you need to pay attention to is the potential of the land which will comprise most of the city's working territory. You need tiles with Food potential to feed your citizens, and tiles with Production potential to ensure your city may build things. Also, any Gold Gold potential is welcome, and is very important in the early game.
River tiles are extremely useful when choosing a place to settle. It's not by chance that most of the greatest cities in the world are on rivers - they convey many benefits, which include the following in the game:
- +1 Gold Gold potential for all tiles next to the river (though this bonus is removed in the Brave New World expansion).
- Allows the city to build some nice buildings, like the Water Mill and the Hydro Plant.
- Farms built next to rivers double their Food Food output early in the game - after discovering Civil Service. Farms in other tiles have to wait until much later (i.e. the Industrial Era) to do this.
Coastal cities also enjoy similar benefits, allowing construction of bonus sea buildings like the Lighthouse. Try to ensure access to some sea resources, too.
Strategic Advantage
Finally, when choosing places to settle, consider military matters. Settling a city on hills will give it a nice defensive bonus (which directly increases city Combat Strength, so it's actually both a defensive and offensive bonus). But more importantly, observe where your neighbors are trying to settle, and try to block their territorial advancement by founding cities on your own. This may lead to diplomatic protests, but it's better than risking your empire's progress and expansion being stymied and stifled by your neighbors.
It's a good idea to carefully observe the most likely places for settlement between your territory and your closest neighbor's territory, then settle there first. Even if they're farther away from your capital, you'll gain a tactical advantage by taking over the contested territory and any resources in it before your neighbor does. Later, you can settle places closer to the capital at your leisure.
Roads
When building a city, keep in mind that you'll eventually want to connect it to the road network leading to your capital. Creating a road (or railroad) not only helps sustain your empire, but also allows quicker transport for Workers to build Farms, Mines, and other improvements. When close to a (future) enemy, defensive and offensive units can take positions much sooner. If you are playing as Rome and have a Legion or two, you can use them to quickly build a road between your new city and your capital.