City States are the one-off cities in a game (Auckland, Genva, La Venta, etc). And you can earn envoys through various tasks, policies, cultures, etc.
These envoys then can be assigned during the City State elections where all civs assign their envoys.
To get the achievement you need to win out and have the most envoy's for a city state so they support you the most and make you the Suzerian. On the Envoy Screen it will tell you how many delegates you need to obtain Suzerian status.
The best bet is to focus and put all envoy's into one state close to you to try and gain Suzerian status.
You can help this by building trade routes, completing their quests, or some policies can help give you extra boosts to their status.
The achievement will pop after you've achieved the status after a delegation turn.