Before the 1840s national currency was more or less non-existent, like the US dollar, was rare and only used in big northern cities, like Philadelphia and New York. Early dollars are so rare they are collectors items.
In virtually all the states a local currency was used, which in North Carolina was called the "pound". Technically the pound was denominated in US dollars, but in reality it was denominated in Spanish silver dollars which were also used as currency and were directly convertible into the local currency. So, for example, you could pay in Spanish dollars at a fixed rate of exchange which was considered equivalent to the pound.
The other option was to get a bank note. This would be issued by a private bank and would be denominated in "pounds". In the South, the bank itself would be backed by Spanish dollars. So, your money would say something like "Bank of Charlotte, 10 pounds, signed [Treasurer of Bank of Charlotte]". If you took your "bank note" to the bank and demanded specie, they would give you Spanish dollars.
In the North, things were different. In Boston, where I live, we used what we called "country money" or "Boston money". These were bank notes, just like those in North Carolina, but the difference is that we did not have much access to Spanish dollars. For this reason, it was difficult to get specie and everything was done in bank notes, unlike in the South where specie was much more common. There were actually quasi-laws that basically made it more or less impossible for the average person in Boston to get specie from a bank.
If you were rich and connected, you could demand specie in Boston under the right conditions. If you did so, what you would get would be some kind of foreign money, probably either English, French, Portuguese or Spanish. I quote from a law of 1806 which gives you an idea of what you might get:
Be it enacted...[blah blah]... a legal tender for all debts and
demands at the several respective rates following and not otherwise:
The gold coins of Great Britain and Portugal at the rate of one
hundred cents for every twenty seven grains of the actual weight
thereof; the gold coins of France, Spain, and the dominions of Spain
at the rate of 100 cents for every twenty-seven grains and two fifths
of a grain. Spanish milled dollars at the rate of 100 cents each, the
actual weight whereof shall not be less than 17 pennyweights and 7
grains, and in proportion for the parts of a dollar; crowns of France
at the rate of 110 cents for each crown, the actual weight whereof
shall not be less than 18 pennyweights, 17 grains, an in proportion
for the parts of a crown."
-- US Currency Act of 1806
Of course, the key thing here is that this was a United States law, not a Boston law, which means it was pretty much ignored. What the bank would usually do is discount your bill, so you might only get 24 grains instead of 27 grains or whatever. And what you get could be anything, French crown, British Guinea, Portuguese escudo, anything. Not surprisingly, most Bostonians just gave up and did everything in bank notes.
By the way, the New York banks heavily discounted Boston money, which meant that you basically lost a LOT of money if you tried to take it out of Boston.