The us has a very messy currency system...and it is designed to give maximum control over the us economy. I can understand why you are confused by it.
The federal reserve LOANS money to private banks. The interest they charge sets the federal interest rate. Banks that want to make a profit are obviously forced to charge a higher interest on their loans than the federal rate.
The federal reserve also does not print the actual money. Us engraving and printing does.
The federal reserve is sort of private...but the pres appoints the chair...the house sets the salary...and the profit of the federal reserve is taxed by the federal government at a rate around 96%