This article is a little outdated, especially in the section related to Influence points flow. Read it with caution, because it might not correspond completely to the current state of the game (Brave New World expansion).
Influence is the main system of determining how a City-State feels towards you or a civilization. Influence is determined in many ways. Hostility, such as demanding tribute, passing within the City-State's borders without permission, or declaring war on them, will lose you influence points. However, kindness, such as gifting units or gold to City-States, completing missions they make, or planting spies in their city, will end up in you gaining influence points. There are five stances a civilization can have with a City-State (in order from worst to best): Permanent War, Angry, Neutral, Friends, or Allies. There is also the status "Afraid", introduced in Gods & Kings.
Resting Points, Influence Points Deterioration and Personalities
Your relations with a City-State are determined according to an Influence scale, which goes from -60 to + X (where X is a currently unknown number). Because City-States tend to be forgetful, effects of actions that have altered relations will subside with time, which is shown by a natural 'drifting' towards a 'Resting Point', which means that each turn Influence points will change naturally (without you doing anything), going towards that point - either increasing, if they were below the point, or (as usually happens) decreasing, if they were above it.
The Resting Point(s) of a City-State is the number of Influence Points at which they cease changing by themselves, a 'natural 0' point in your relations with that State. While by default this is 0, this may be altered by the Religious Belief "Papal Primacy" (+15 Resting Point for all City-States following this religion), the Patronage Social Policy "Aesthetics" (in vanilla) or "Consulates" (+20 Resting Point for all City-States), or by pledging to protect a City-State (+5 Resting Point for the City-State you are pledging to protect). Pledging to Protect a City-State will announce to the world that if they want to get to the City-State, they're going to have to get through you. This acts as, more or less, a Defensive Pact between you and the City-State.
With every City-State comes their Personality. City-State's Personalities are predetermined for the specific City-State, and will not change even if you enable Random Personalities. Personalities can be Neutral, Friendly, Hostile, or Irrational, and their main effect is the rate at which Influence points change naturally. Personality also involves what kinds of special quests they will give you.
Neutral City-States lose positive Influence Points and regain positive Influence Points at the same rate. The rate is, by default, 1.00 Influence Points per turn, but can be altered by the unique ability of the Greeks and some Patronage Social Policies. Neutral City-States usually mind their own business and may not give you too many quests at a time.
Friendly City-States lose positive Influence Points at a slower rate and regain positive Influence Points at a faster rate. They will give you quests like building wonders, connecting them to one of your cities by road, or finding another civilization's land. When you want to befriend City-States, these are the ones to look out for.
Hostile City-States lose positive Influence Points at a faster rate and regain positive Influence Points at a slower rate. They simply don't like other City-States or civilizations; befriending them for long is difficult without Social Policies or other abilities that alter the rate of which Influence Points are changed. Hostile City-States tend to give you quests to eliminate (conquer) or bully (demand tribute from) another City-State.
Irrational City-States lose positive Influence Points and regain positive Influence Points at a faster rate than normal. They will give you more quests than City-States with other personalities. As such, they can become potential Allies, just keep doing their quests, because they also lose Influence as if they are Hostile. But if they are wary, they regain Influence as if they are Friendly. Use this advantage to restore relations with these City-States should they be angry at you.
Friends and Allies
When you are either Friends or Allies with a City-State, you can get a variety of benefits. How beneficial these are based on whether you're Friends or Allies with them.
- Friends - attained at 30 Influence Points. If you are Friends with a City-State, they will give you unique gifts based on their type: Culture for Cultured City-States, Food for Maritime City-States, Units for Militaristic City-States, Unique Luxury Resources for Mercantile City-States, and Faith for Religious City-States. They also grant you the right to use their territory as though it were yours, although you can't construct any improvements there except for roads, railroads, and, if Portugal, Feitorias.
Note that you can be Friends with as many City-States as you can influence.
- Allies - attained at 60 Influence Points. If you are Allies with a City-State, they will not only give you more of the previously mentioned gifts, but they will also give you access to all of their resources - both luxury and strategic. This stands in for the lack of trading for City-States to have a way to still acquire their resources without having to conquer them. You can use the resources in your Empire, as though they were yours, but you can't trade them away! Allied City-States also share the vision of all their units and territory with you.
Whenever you are at war with another civilization, City-State allies will also declare war on whoever is at war with you. However, only Militaristic City-States will provide you units while you are at war.
In Brave New World, City-State allies also provide additional Delegates for you to use in the World Congress.
Note that each City-State can be Allies with only one civilization (which is also called their Patron). If more than one civilization has 60+ Influence with a City-State, it'll ally itself with whomever has more Influence - hence the unlimited upper border of the Influence scale.
Afraid, Angry, and Permanent War
If you don't feel like being very friendly to your local City-States, there are three other stances: Afraid, Angry, and Permanent War. They happen when your Influence with them drops into negative territory. Unlike their Friends and Allies counterparts, they will not provide benefits for your civilization unless you conquer them.
City-States are Afraid of you if you have a military force that outmatches theirs and you are close to them (within 8 tiles of the city). Afraid City-States can be asked for tribute and will always comply to giving anything you want. When asking for tribute, you can either demand gold from them or enslave one of their workers. This is useful early-game when you have a strong military but are in need of money or workers. However, do note that either of the previously mentioned actions WILL still cause you to lose influence with them, and doing so will often cause them to change their stance from "Afraid" to "Angry". Stats regarding how scared a City-State is can be seen by mousing over the options in the "Ask for Tribute" menu. These stats include total military might, nearby military units, if tribute was recently demanded, and if the City-State has a pledge of protection with another Civ.
City-States are Angry at you if you have done something they strongly dislike. Common actions include trespassing on their borders, demanding tribute, or declaring war on them. Other, less common incidents, such as a civilization that is allied with a City-State declaring war on you, will cause them to get angry at you (or, in the last mentioned case, declare war on you). Angry City-States require more kind actions to please them. This is the worst state you can be with a City-State until you reach Permanent War (see below). A City-State is Angry at you when your Influence Points with them are negative. Note that War with a City-State is achieved with -60 Influence Points with the City-State.
Permanent War is the worst state you can be with a City-State. Permanent War is achieved when you have conquered, declared war on, or killed the units of so many City-States that most City-State will believe that any further peaceful coexistence is useless with you. In this case, you will enter Permanent War with most, if not all, City-States. As the name suggests, this is a form of war with a City-State where peace cannot be established. This means you will forever be at war with the City-State(s) -- at least, until you conquer them.
When a City-State is Neutral , nothing major has happened in your relationship that has caused them to like or dislike you. Neutral City-States provide nothing good or bad to you. A City-State is Neutral when your Influence Points with them are 0-29.