There are 4 different personalities that I've seen for city-states: friendly, neutral, hostile, and irrational. I have never seen any use for them, though, except that influence over hostile city-states decreases at 1.5 times the rate of the others. What do the other personalities mean?
It affects the kinds of requests the city-state makes. From the game's documentation:
A befriended Friendly City-State will usually make requests that require some peaceful action, like building a Wonder or constructing a Road to them.
Neutral City-States are fairly even-keel, and try to mind their own business.
Influence with a Hostile City-State drops more quickly than normal. They will also make requests to capture neighboring City-States more often.
Compared with other City-States, an Irrational City-State is more random with the requests it makes.