The first CENTRAL bank of the US was the Bank of the United States, and a wikipedia article is linked by the first guy that answered you. But that was NOT the first bank in the US, not by a long shot.
The first chartered bank in the US was Philadelphia's Bank of North America, chartered in December 1781 by Congress, and opened for business in 1782. This was the first US chartered bank that ordinary people like you and me could open accounts at. The bank was located at the corner of 6th and Chestnut, not far from Independance Hall, and that bank branch has been in continual operation since then to this very day, making it the oldest continously operating bank branch in the US. It is now a Wachovia Bank branch, and even today you can go there and open an account!
Bank of North America is an "ancestor," or predecessor bank, of Wachovia Bank, NA, and to this day Wachovia Bank, NA, has th e original National Bank Charter Number 1 given in 1781, which it inherited from the Bank of North America. Go to the website of the Comptroller of the Currency, and you can look up national banks by their charter number. Look up number 1, and Wachovia Bank pops up.
Wachovia just agreed to merge with Wells Fargo. Depending on how Wells Fargo handle the merger, National Bank Charter Number 1 may finally die soon, or Wells Fargo & Co could instead merge Wells Fargo Bank, NA, into Wachovia Bank, NA, and therefore effectively get Charter Number 1. It can then just change the name of Wachovia Bank, NA, to Wells Fargo Bank, NA, so that both the Wells Fargo name and Charter Number 1 survive the merger. I guess we will see how they handle it.