The civilization with the most Envoys at a city-state, and at least 3 Envoys, is considered that city-state's Suzerain. There can be only one Suzerain; ties are not allowed. If two or more civilizations have the same number of Envoys at a city-state, there is no Suzerain.
The Suzerain of a city-state gets several significant benefits.
Unique city-state bonus
This is a powerful bonus which no other city-state will provide. For example, Buenos Aires will cause Bonus Resources to act as Luxuries and provide +1 Amenity per type, while Rapa Nui will allow the construction of the Moai, a unique tile improvement.
Inside the city-state's borders your units are "at home." They can heal there at an accelerated rate and be upgraded.
The city-state will automatically follow its Suzerain into war and peace, matching the diplomatic state of the Suzerain. In Gathering Storm it will also provide +1 Diplomatic Favor per turn to its Suzerain.
If the city-state gets attacked by a third party, its Suzerain will immediately gain the Protectorate War Casus Belli against the aggressor. In Gathering Storm, the Suzerain will also gain Grievances against it.
All of the city-state's resources are automatically exported to the Suzerain, and the Suzerain may build tile improvements in their city-state's territory. Note that the Suzerain may make use of these resources, but may not (in the case of Luxuries) trade them to third parties.
The Suzerain may also pay Gold to levy (i.e. take control of) the city-state's military units for 30 turns. The amount of Gold that needs to be paid is equal to the total Production cost of all military units the city-state owns.
Kilwa Kisiwani, a Medieval Era wonder in Rise and Fall, gives its owner a 15% yield bonus from any city-state of which he or she is the Suzerain (and an additional 15% yield bonus for being the Suzerain of more than one city-state of the same type).
New Orleans Style Spanish Rice
Become Suzerain of a City-State.