To unlock co-op mode in Genshin Impact, you need to reach Adventure Rank 16. Once it is unlocked, you can see the multiplayer button in the main menu. You can also access multiplayer by entering Domains, which are basically Genshin Impact’s version of Dungeons which unlock later.
In Genshin Impact co-op mode, you can basically do many of the same things in single player. You can join other people’s games or host one yourself. However, each session will have someone as the main host. During a session, you can go around exploring and doing quests just like in single player.
However, there are some things you cannot do in multiplayer. You cannot go on story quests, as they are all paused when you go multiplayer. You also cannot collect Geoculus or Anemoculus, open chests, or make offerings to the statues. Only host players can do this.
Hope this helps.