Civ 5 - how to protect city states?

You can pledge to protect a City-State. Before the Gods & Kings expansion the game required you to at least be friends with the city state before you are allowed. The mechanic changed for the expansion however, now allowing you to pledge to protect city states immediately. For some reason, the AI is allowed to pledge city states right away, even in the base game. G&K just evened the playing field in this sense, but you won't need to buy the expansion to use the mechanic.

It's in the detail window for a city-state, second option down - Pledge to Protect.

Note you cannot pledge to protect a city-state if you are at war with them.