It looks like there's a risk that this question will go unanswered. Just in case that happens, this link and this link have a pretty good overview of the evolution of banking in the US.

The short answer is that over your time period the US went from local banks (that generally needed a State charter, though their was no national requirement for a State charter) that took depositors money and issued their own notes, to the current system guaranteed and overseen by the Federal Reserve.

Your times don't really match up to the evolution of banking in the US. Basically banking would be classified as follows:

1. Pre-1791: banks are regulated by their states and branches (though not necessarily lending) are confined to a given state. Banks can issue their own notes, and notes from other banks are typically redeemable at other banks (say, in a different state) at a discount. This creates a problem with excess credit (as banks issue notes, leading to inflation), hampers inter-state trade (different currencies, or limited to gold), and hampers inter-state credit flows (so excess capital in the East is harder to move to growing communities in the West).

2. 1791-1836: The First/Second National Bank. The Federal Government gets involved, through a private bank of which the government is the largest shareholder. The National Bank was doomed politically when it responded poorly to the crash of 1819 (it had facilitated too much lending to agricultural land, that lost a lot of value as agricultural prices fell as Europe recovered from the Napoleonic period; and then overreacted, tightening credit and making the resulting depression worse in the name of fiscal restraint).

3. Pre-Civil War. Back to State banks, also called the "Free Bank" period, because of the lack of regulations.

4. Civil War: 1863 back to National Bank, and by 1865, a National Currency. More regulation on currency and interstate lending, but in stark contrast to the modern era, banks had no insurance: if they lent too much, they could go bankrupt if depositors demanded their money all at once. As the whole point of banks is to take depositors money and LEND it out, this made banks very vulnerable to panics. In times of crisis, a depositor had little to gain by not withdrawing money, and everything to lose. Ironically, the governments hesitance to guarantee banks (fearing moral hazard and corruption), made banking much less stable. There was a panic every decade.

In contrast, by the 1870's, Britain had a central bank that would lend to component banks during a crisis to keep them afloat. Bagehot's famous line is "lend freely at a high rate of interest against good collateral".

5. 1913: The Fed. Starting with emergency lending to combat bank panics, and regulations to control credit bubbles and facilitate national lending. By 1920, the Fed started to actively conduct monetary policy (controlling the amount of money in the economy to smooth out booms+busts), and the importance of this aspect picked up markedly after the US left the gold standard in 1933. Since then, while the economic theory underpinning central bank operations has greatly evolved, the structure has stayed the same: 12 regional reserve banks (one change: they don't each print money anymore) and a 7 person board of governors nominated by the President and confirmed by the Senate for 14 year terms.