What is the difference between an emby, a legation and a mission ?

"Emby" is now the normal title for the main representative office of one country to another (or to an international organization. An Emby is headed by an Ambador. A Legation is now quite rare, it was a lower type of representation, headed by a "Minister" they are quite rare nowadays since most countries maintain embies in other countries. A "mission" is a generic term for the representation of one country in another and includes embies, legations and consulates. Embassies to international organizations are often referred to as missions.