system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: What is the Federal Reserve and why do so many people hate it?
gpt
The Federal Reserve is the independent central bank of the United States.  It is controlled by a Chairman and twelve members of a board of governors (one for each District Reserve bank) who are appointed by but not controlled by the President (thus "independent" central bank).  One major criticism leveled by Ron Paul and others is the idea that the Fed is "unelected", but this is actually by design; a dependent central bank faces a moral hazard from the other parts of the government.  More on that later.

It is responsible for the supply of money by controlling what's known as the Discount Window, or the lender of last resort - essentially, it loans money to banks - and though Open Market Operations, which are the sale and purchase of U.S. Treasury bonds.  It also sets the Reserve Rate, or the ratio of assets to obligations that a bank must maintain to stay solvent (a bank needs to have enough cash on hand to pay out daily operations, but it wants to have as much of its assets loaned out to other people - the Fed's requirement means that a bank can't be overly aggressive with this ratio).

The Discount Window is a hyper-short-term (usually overnight) lending facility that banks can use as a last resort, with an interest rate that targets a certain "supply of money" (the amount of liquid currency in circulation), which influences the Federal Funds Rate, a standard interest rate that banks charge each other for overnight loans.  Banks need these overnight loans because daily operations are not stable - one bank may have more assets-to-obligations than required, while another may be below the requirement...so overnight loans are used to make up the difference.  If the FFR is higher than the Discount Rate, nobody would use it; banks tend to lend to each other and then use the Discount Window if they can't find a loan at the lower rate.

The Fed also influences the FFR by Open Market Operations - the buying or sale of U.S. Treasuries - as these will provide banks (the typical customer) with a flexible yield (the customers compete for the rate, thus "open market").  When the FFR, which is monitored by Fed officials, goes outside a window, somebody at the Fed building in DC literally pushes a button to sell or buy a bond.  This removes (sell) or adds (buy) money to the banking system.

Why does the Fed target the FFR? Because it influences two important numbers that the Fed is supposed to be watching: unemployment and inflation.  Markets prefer low (positive) inflation rates and low unemployment, though economists believe in a "natural" rate of unemployment where inflation is 0 (the NAIRU - NonAccelerating Inflation Rate of Unemployment) that comes as people quit or are fired.  Governments running deficits prefer high inflation, as it devalues their debt.  This is the reason that the independence of a Central Bank is important!

Inflation has two components - expected inflation and monetary policy.  Inflation expectations cause inflation, as market participants looking forward to more inflation will hike their prices now.  The Fed, however, is the only entity with knowledge of actual monetary policy at the time it is happening (markets discover this information eventually).  This means that the Fed can run monetary policy below or above expected inflation, and from there influence the unemployment rate (inflation of money supply > expected inflation = more money in the system than people think there is, which gooses growth in the short term)...even below the natural rate of unemployment!  Eventually they get caught, and expectations rise (which eliminates the benefit and actually makes it worse, because inflation is now higher).  So the Fed tries its best to make the markets think that inflation will be low, and the best way to do that is to keep actual inflation low (even if that causes a rise in unemployment rates).

Current FFR policies are extremely loose (due to the high unemployment rate and liquidity crisis), but inflation has not followed suit.  This leads to charges by Fed-haters that the government is flooding the market with money and "monetizing debt" - financing deficit spending by the fiscal side of the government.

I've explained the neoclassical economics (with a dash of Friedman and Keynes) view of monetary theory.  Another, very complicated, model used by the Austrian school of economics tends to posit that the central bank is counterproductive due to business cycles.  Most other criticisms of the bank (Unconstitutional, tool of the Illuminati, etc) tend to be conspiracy theory based.