I'm not a historian. But I worked for three years as a research assistant with an economist who studied this specific time in history and recently published in the Am Econ Rev on it. (Our project specifically dealt with correspondent relationships between these banks.) I'm currently working at a federal bank regulator's research division.

The period in history you're talking about is called the "free banking era" (also the "state banking era") and was dominated by the use (and at times abuse) of banknotes. My answer assumes you are familiar with M1, M2, etc definitions of money.

What are banknotes?

Specie, ie gold and silver, was the widely recognised "cold hard cash" of the day, usually in the form of coins. But the notes which most people actually used and which made up most of the money supply were not specie. To get a bank note, you give a bank something valuable (land, specie, government bonds, etc) and receive a piece of paper saying they will give you a dollar if you return it. Mutatis mutandis for quantity etc. Eg, I have a 1990 fiver on me right now, proclaiming "I promise to pay the bearer on demand the sum of five pounds" above a depiction of Britannia and the words "For the Governor and Company of the Bank of England" and signature.

These phrases are a legacy of the origin of the banknote itself. To summarise a centuries-long series of development very shortly, people with gold or silver put their stuff in a vault. Then vault-owners gave them a receipt. Then they said "how about we made this tradeable so you don't have to actually go to the vault to get the coins". The receipt for depositing about an ounce of silver then becomes just a bearer instrument on it and you buy stuff by trading the receipt.

This implicitly had a limit on how much a bank (operating legally) could issue. If a bank issued too many banknotes, then people might redeem them and if the bank does not have enough specie or time to get specie, it goes under. States also imposed required reserve ratios: if you have 1000 dollars worth of notes, you need to have on hand enough specie to pay off, say, 250 dollars worth of those notes.

Commerce and regulation

There are a few issues of course, with this system. The largest perhaps is that banks can fail. Moreover, exchanging notes is annoying. Also, because the US is made up of a ton of different states, there are no consistent safety and soundness requirements. I'll explain these in reverse.

Charters. First, a bank is a company which is chartered by some government. Early in the American republic, there was a national bank chartered by Congress. The Second Bank of the United States' charter expired in 1837, meaning that all banks then rested on their state charters. The state charters also meant that a bank in New Jersey usually could not set up a branch in New York (unless the two states had an agreement allowing interstate branching; this restriction was only lifted nationwide in 1994).

Discounts. Exchanging notes is also annoying. If you come in with a note from a bank that claims it is in some place called "Chicago" in Illinois and I, living in Philadelphia, have never heard of it and will likely never go there to claim the dollar that it supposedly would allow me to claim (even if it isn't counterfeit), I am going to laugh at you and tell you to return with real money. You could go, however, to a bank or a broker and exchange your note for specie, with some major discounts below par, depending on published rates. Eg notes from Illinois in 1845 were accepted with a 70 per cent discount in Philadelphia. If your note was from a well-known New York bank (eg the Bank of New York, now BNY Mellon) then the discount would be more like 0.5 per cent. Economists would describe this as trading at a discount (or below par).

If you had a note from a reputable bank which people have heard of, it likely meant you were in that bank's locality. Banks in this time period and through to the 1930s also printed their vital statistics (especially their capital) in newspapers to advertise that putting money there would be safe. To move your money from Philly to New York, you needed to get on the interbank "network" (automated clearing house or ACH is the spiritual successor). Instead of exchanging all my money for silver, and then probably getting robbed while travelling, I make sure my bank has a correspondent account in New York with an agreement that the NY bank honours my notes. Then, when I go to NY they take my notes with my Philly bank name, give me the coins I need, and settles them bank-to-bank at the clearinghouse.

New York's clearinghouse was especially large. There were clearinghouses in every general region before sending the interregional aggregated transactions to New York. A small bank in a rural area might need to go through multiple levels of clearinghouses, unless it had a direct correspondent relationship.