CONDUCT DIPLOMACY
After you've decided which diplomatic arrangement you'd like, contact the foreign country by looking up their email address in the Rolodex.

In order for a diplomatic arrangement to become active, the Administrator must receive notification (usually in the form of email) from you and more importantly, independent notification from the foreign country as well.

SECRET DIPLOMACY EXISTS TOO - JUST DON'T TELL THE ADMINISTRATOR!

Listed below are some typical diplomatic arrangements.
Feel free to make up your own.

BILATERAL ALLIANCE: Self-explanatory, remember to set the terms yourself (e.g. Entente Cordiale between Britain and France).
MULTILATERAL ALLIANCE: Self-explanatory, remember to set the terms yourself (e.g. NATO).
PROTECTORSHIP: One country (the 'protector') controls the foreign affairs of another (the 'protectorate') in return for protecting that country (e.g. late 19th century Egypt by Britain).
MANDATESHIP: One country controls the foreign and internal affairs of another in return for protecting that country (e.g. Syria by France as assigned by the League of Nations).
SPHERE OF INFLUENCE: An agreement between two countries that one of those countries has dibs on another country that is not currently controlled by either of them (e.g. Britain and Russia going after Afghanistan in the Great Game).