embassy
Refers to the diplomatic mission building or the institution itself. Often used with location particles に/で.
I went to the embassy to renew my passport.
The embassy is open from 9 a.m.
Japan's embassy is in London.
I went to the embassy to get my passport reissued.
I applied for a visa at the American embassy to study abroad.
If you run into trouble abroad, first consult the embassy.
大使館 refers to an embassy as an institution or office, while 公館 can denote an official's residence itself, though the two overlap in diplomatic contexts like 在外公館.
外務省 is Japan's central ministry overseeing all foreign policy, whereas 大使館 is a single embassy representing Japan (or another country) abroad.