Fractional Reserve Banking is an interesting subject to say the least but depending on your basis of economic theory beliefs, it's everything from Satan's Spawn to Manna from Heaven. UPS Lifer has it correct that' the Fed is a quasi public/private institution but understanding it's nature can be a daunting task at times. I've read a lot over the reads on the Fed from books advocating all sorts of wild conspiracies to more mundane works and of them all I would reccommend 2 that I'm not even sure they are still in print as they were piblications of the Fed themselves back in the late 70's and early 80's. They are entitled
"Modern Money Mechanics" and if I remember correctly (don't quote me on this one because it's been a long time) was published by the Chicago Fed. The other from the NY Fed (again same warning about actual source as time and age plays tricks)has a rather eye catching name in, "I Bet You Thought!"
Both publications were very good at explaining a lot of how the Fed works and yeah to a lot of folks at the time they made some amazing admissions about out banking system but having visited the Atlanta Fed myself years ago and talked with the nice folks there during a tour (I highly reccomend any Fed Bank tour if you ever get the chance) what we are surprised to learn, they think nothing of it and are shocked to learn that we don't already know this stuff. In fact, their openness of actually how the Fed and our banking system works from their own lips IMO dispells a lot of the conspiracy theories running around out there.
If you really want to understand a major problem of money and economics in America and across the globe, I'd highly reccomend you study the 1944' United Nations Monetary and Financial Conference otherwise know by the name Bretton Woods Agreement as the conference was held in Bretton Woods NH. IMO this global centralization of economics had many far reaching consequences on multiple fronts and would be a better use of your time and effort that to point at the Fed as some monsterous boogie man that the vast majority of folks don't understand it function to begin with.
Are you going to now delve into the Jeykll Island Georgia meeting and the actions of Sen. Carter Glass and Paul Warburg doing the bidding of the Bank of England and the Rothschild family?