An embassy is a building or compound utilized by one government, within the territory of another, for the purposes of diplomacy. Embassies are generally in capital cities, while smaller consulates are found in other locations. Embassies and consulates are also referred to as diplomatic missions, and are considered the sovereign territory of their operating government, regardless of their location. Thus, an individual traveling in foreign territory may take refuge at their government's embassy or consulate.
Many worlds and galactic powers have embassies on Earth, and the United Federation of Planets maintains embassies and consulates on its member worlds and colonies and those powers and planets with which it maintains diplomatic relations.
Jean Sinclair was assigned to the Federation Embassy on Andor in 2362, and again (temporarily) in 2364. (Star Trek: Pendragon: "Wings As Eagles")
Many Federation starbases, especially highly-trafficked ones like Deep Space 9, have consulates.
During the 22nd century, the Vulcans maintained consulates in Berlin, Canberra and San Francisco. Likewise, the United Earth had an embassy in ShiKahr on Vulcan, which was bombed in 2154. (ENT: "Fusion", "Terra Prime", "The Forge")
The Klingon Empire maintained an embassy on the neutral planet Farius Prime, which was the target of an unsuccessful break-in by the Orion Syndicate in 2374. (DS9: "Honor Among Thieves")
The Federation, Klingon Empire and Romulan Star Empire all established embassies on Kal-Dixas in 2370. (Kal-Dixas Spaceport)