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:
Unique Bonus: This is a powerful bonus awarded to the Suzerain which no other city-state will provide.
Suzerain Diplomacy Bonus:
The city-state follows you into war and peace with other players.
City-state lands become friendly territory. Your units can move there, heal there, and upgrade there.
Your Builder units may improve city-state tiles.
See 3 tiles around the city-state.
Can pay Gold to temporarily take control of the city-state's military.
Gain ownership of all the city-state's resources.