A city-state is a city that stands on its own. It isn't part of a bigger civ; it is its own civilization. They can build armies, workers, etc. but they do not take active parts in politics/diplo.
You can use money to buy influence with city states. High influence with a CSs will make them your allies, and they will give you stuff. There are three kinds of CSs - maritime, cultural, and militarist, and each give a different kind of gift. Maritimes give food to your cities, cultural give culture, and militarist often give you military units.
You can declare war on an conquer CSs as well.