Delegations and Resident Embassies can be established between players to improve relations and gain Diplomatic Visibility.
A Delegation, or Embassy, needs to be sent to every leader individually. Delegations cost 10 Gold and Embassies cost 25 Gold, which is paid to the other leader. This will give you a small positive bonus in your relationship with that leader. It also gives you 1 level of Diplomatic Visibility. See the Diplomatic Visibility and Gossip section of the Civilopedia for more information on Diplomatic Visibility.
Resident Embassies are unlocked with the Diplomatic Service Civic, at which point they replace Delegations. The effects of Resident Embassies and Delegations do not stack.