What kind of banks existed in the US before the National Bank was established by Alexander Hamilton?

Bank of NY was established before the Bank of the US. It was a strong competitive entity. Other states also had banks of sorts but not necessarily official state banks. Can't help you with names, when established, or how long those lasted except to say one did not obliterate the others. Problem with not having a federal level bank was that we remained dependent on Congress to bring the 13 States into agreement for funding all the Revolutionary War debt that Hamilton had managed to bring under Treasury auspices.

The Bank of the US that Hamilton created was what provided the first real financial stability and prosperity to all classes in our nation. Just as important to that end he helped in establishing the Society for the Encouragement of Useful Manufacturers (SUM) - essentially a venture capital corporation - to promote domestic production and thus independence from Europe. Meanwhile Jefferson, Madison, and Monroe were enemies as they wanted to maintain the agrarian base and the landed gentry as rulers. These were deep seated disagreements filled with rancor. A story at the end of "Alexander Hamilton" by Forrest McDonald makes this clear.

Many years after Monroe's presidency he called on Betsy, Hamilton's widow. when she entered the parlor he rose. She remained standing facing him in the center of the room. She did not ask him to sit down. He stumbled through saying "it had been many years since they had met, that the lapse of time had brought its softening influences, that they were both nearing the grave, when past differences could be forgiven and forgotten." She replied "Mr. Monroe, if you have come to tell me you repent, that you are sorry, very sorry, for the misrepresentations and the slanders, and the stories you circulated against my dear husband, if you have come to tell me this, I understand it. But otherwise, no lapse of time, no nearness to the grave, makes any difference."

They were local banks and state banks. When the National Bank took over the local and state banks no longer existed.

Up until Hamilton there were only local banks that circulated their own notes of credit. One of the major reasons that a strong central government was necessary was for a single currency in the country. Before then each state in fact each community was circulating there own notes of questionable value. There was also no way to control a panic, some one started a rumour and everyone showed up at he banks front door the next day to withdraw their money. There was no central clearing house either. If you had to cash a check from NYC Bank in Boston chances are they wouldn't honor it because there was no way to get their money back.

Many, many local banks existed, and they continued to exist after the Bank of the United States was established. Most of these local banks were totally unregulated, and issued currency that was often not very safe because it wasn't always backed by actual deposits in the bank. The large number of types of local paper money in circulation because of these banks led to a huge counterfeiting industry in the early USA.

See the book "A Nation of Counterfeiters" by Stephen Mihm for some fascinating and often hilarious stories about this.