A diplomatic mission is a group of people from one state or an international inter-governmental organization (such as the United Nations) present in another state to represent the sending state/organization in the receiving state. In practice, a diplomatic mission usually denotes the permanent mission, namely the office of a country's diplomatic representatives in the capital city of another country.
A permanent diplomatic mission is usually known as an embassy, and the head of the mission is known as an ambassador. Missions between Commonwealth countries are known as High Commissions and their heads are High Commissioners (although many 'New Commonwealth' countries choose the more common terminology in dealing with other New Commonwealth countries; in New Delhi for example there is a Canadian High Commission and a Singaporean Embassy). All missions to the United Nations are known simply as Permanent Missions, and the head of such a mission is typically both a Permanent Representative and an ambassador. Some countries have more particular naming for their missions and staff: a Vatican mission is headed by a Nuncio and consequently known as an Apostolic Nunciature, while Libya's missions were for a long time known as People's Bureaus and the head of the mission was a Secretary. (Libya has since switched back to standard nomenclature.)
In the past a diplomatic mission headed by a lower ranking official (an envoy or minister resident) was known as a legation. Since the ranks of envoy and minister resident are effectively obsolete, the designation of legation is no longer used today. (See diplomatic rank.)
In cases of dispute, it is not uncommon for a country to recall its head of mission as a sign of its displeasure. This is less drastic than cutting diplomatic relations completely, and the mission will still continue operating more or less normally, but it will now be headed by a chargé d'affaires who may have limited powers. Note that for the period of succession between two heads of missions, a chargé d'affaires ad interim may be appointed as caretaker; this does not imply any hostility to the host country.
A Consulate is similar to (but not the same as) a diplomatic office, but with focus on dealing with individual persons and businesses, as defined by the Vienna Convention on Consular Relations. A Consulate is generally a representative of the Embassy in locales outside of the capital city. For instance, The British Embassy to the United States is in Washington, D.C., and there are British Consulates in Los Angeles, New York City, Houston, and so on.
The term "embassy" is often used to refer to the building or compound housing an ambassador's offices and staff. Technically, "embassy" refers to the diplomatic delegation itself, while the office building in which they work is known as a chancery, but this distinction is rarely used in practice. Ambassadors reside in ambassadorial residences, which enjoy the same rights as missions.
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