An embassy, also referred to as a permanent diplomatic mission, are the representative body of one state or territory to or within another. Embassies represent their country or state abroad in diplomatic interests, as well as the interests of its citizens that may be living, working, or visiting the host country. They should not be confused with consulates, which are akin to satellite offices and under the authority of the embassy. The head of the embassy is called the ambassador.
To be completely technical though, an embassy is defined as the diplomatic delegation itself, though most people harmlessly call the main physical building as such (the name for the building is actually a chancery).