Civ V City-States - The Basics
The simplest way of describing Civ V city-states is as small civilizations. As you set out to explore from your first city, you will encounter city-states that are much like other civilizations, but only have one city and are much less aggressive. In the early game, they will give you gold for encountering them and may make requests that you help them with barbarians. Even if they survive to the very end of the game, city-states will never grow beyond one city, though their borders will expand. They may seem insignificant at first, but in truth, they are essential to victory.
City-states will, when friendly or allied, provide your civilization with gifts of culture, food or military units. You gain these bonuses by accruing influence with city-states, which you can get by gifting gold to them or helping them with other problems. This influence will decay by -1 a turn, so it can be difficult to maintain multiple allies. City-states usually have access to luxury and strategic resources as well, which they will share with you if you become allies. If managed correctly, city-states can boost your civilization’s growth massively, help you build an army or assist you in achieving policy goals much faster.
Civ V City-States - Cultured, Maritime, Militaristic
There are three basic types of city-state: Cultured, providing culture, Maritime, providing food, and Militaristic. The culture will, of course, only add to your policy total, but the food from a Maritime city-state will benefit all of your cities, but your capital most of all. The culture and food bonuses do not require a trade route or any connection, so these city-state allies can be very close or very far away from your empire. A militaristic city-state will spawn units for you within their borders, so it is important to keep in mind the location of these allies. There are also different types of personalities - these dictate what kind of requests they will make of you (building wonders, attacking other city-states, acquiring Great People). While most of these are insignificant, be aware that ‘Hostile’ city-states will lose influence faster, so be very careful when allying with them.
Civilization Strategy - City-States and Victory
Early in the game, it can be very expensive to try to win allies with gold alone, so watch for notices from city-states nearby that want barbarians destroyed or help with another civilization. The influence you get from city-state quests is significant, and makes gold gifts more lucrative, since they will boost you to ally status. Maritime city-states can be very effective in the early game, especially if you have enough extra happiness to expand. The Siamese and Greek empires both get significant bonuses to city-state relations (Siam gets bonus resources and Greek influence decays faster) so make an extra effort when playing these civilizations to acquire city-state allies.
A good civilization strategy can also involve conquering city-states, since they often have good locations and valuable resources, but be careful when making your plans. Other civilizations will also ally with city-states, and may go to war with you if you misstep. Also, after you’ve conquered two city-states or more, you will receive a penalty to influence and it will decay faster with the remaining city-states. The new Mongol civilization also receives a bonus to these attacks, so when playing as Genghis, it may still be worthwhile.
Consider also that the Patronage policy tree adds significant bonuses to city-state relations, making it easier to make and keep allies, as well as providing extra resources - science and Great People among them. Combining the Patronage tree with Greek or Siamese civ bonuses can be nearly game breaking.