The civilization with the most envoys on a city-state, and at least 3 envoys, is considered that city-state's Suzerain. There can be at most one Suzerain; ties are not allowed. If nobody has envoys on a city-state, there is no Suzerain. The Suzerain gets several significant benefits, which are described below.
Unique city-state bonus: This is a powerful bonus which no other city-state will provide. For example, Jerusalem acts as a second Holy City for your religion, meaning that Jerusalem will exert religious pressure to nearby cities and city-states.
Diplomatic allegiance: The city-state will automatically follow its Suzerain into war and peace, matching the diplomatic state of the Suzerain.
When at war, the Suzerain may also pay Gold to take control of the city-state's military units for 30 turns. The amount of Gold that needs to be paid is determined by the size and strength of the city-state's military.