The us ambassador at the Vatican (which is a city-state) is the us ambassador to the holy see. It's the formal title for the Vatican.

As I understand it, it can be used 3 ways, similar to the way the word "White House" is used.

1) It's the buildings where the Pope lives, and where the Papal authority, that is the organization that carries out the Pope's role, are housed.

So, for example, I could say. "I've always wanted to visit the Holy See, and enjoy the spectacular artwork that is on display there." Just like I could say "I've always wanted to visit the White House to see . . . ."

2) It's the organization that is housed within the Holy See.

So, for example, you might read "The Holy See announced that the Pope is ill and will not be attending . . . " Just like you might read "The White House announced today that . . . "

3) It can be a way of referring to the Pope himself.

So, for example, you might say "The Holy See sent a proclamation today in support of the actions of . . . " meaning a proclamation that was made by the Pope, in the same way you might say "The X bill was signed at the White House" when what you mean was that Obama signed it. You wouldn't say "The X bill was signed at the White House" if what happened was that Sasha Obama found a draft of the bill, and practiced her signature on it.

"Holy See" means "holy seat" or "holy throne." It's an abstract, ancient term referring to the area of jurisdiction that the Catholic Church (and the Pope as its spiritual leader) has.
The Vatican City is the physical hub of the Catholic Church, and can be seen on a map. But the Holy See refers to all the churches and Catholics around the world that are under the "jurisdiction" of the Catholic Church.
Does that help?

Can someone tell me, briefly and in plain English, what the Holy See is? I looked in Wikipedia but that didn't help much.