I am Puerto Rican - since you guys have wondered in 2 different shows now, I can clarify for you what, exactly, we are
We're officially a Commonwealth, which really doesn't mean much. In practice we're a territory. We have American citizenship and use the dollar, but we don't have senators or representatives in Congress (we have a Resident Commissioner, who is essentially a lobbyist for PR, but he can't vote) and we cannot vote for the president. We can vote in primaries, however, which makes no sense but there you have it. We can also join the military and be drafted (which I know doesn't happen anymore but it has in the past).
Even though we don't have a congressional vote, we pay a variety of federal taxes but not income tax, that one goes to PR's own version of IRS, called Hacienda. In keeping with the limbo we're in, we get some of the benefits states get, but limited or reduced.
We have a governor, mayors, and our own elected officials, along with our own laws & constitution, but it's all ultimately subject to federal law and federal law trumps everything.
tl,dr: PR is a territory and it's complicated, but you can go there without a passport and not have to exchange currency since we're on the dollar.