The Fed does not control the US inter-bank lending rate. It sets a target, and engages in open market operations to try and force the rate to stay close to the target. But it doesn't control the rate.
But these days pretty much the whole term structure of the yield curve is manipulated by central banks (see Operation Twist for instance).
LIBOR (and there are different LIBORs for different currencies and maturities, the main one is USD 3-month) is the rate that banks lend each other on an unsecured basis. Technically, each bank determined their own LIBOR based on how they see the market and what they believe they can borrow at (hence our problems).
The important thing about LIBOR is that many trillions in notional of derivatives use it as a reference, so changing it has very real impact for all market participants (including pension funds, etc.). Even floating rate loans are often based on LIBOR.
Fed Funds are unsecured overnight loans between banks. In the aftermath of 2008, banks have a lot of reserves and trading volume in the Fed Funds market has decreased significantly. It's arguably not an important rate to pay attention to anymore, but the Fed still claims to be targeting it (indirectly through open market operations)
Interest on Reserves is interest paid by the Fed to banks for holding reserves at the Fed. This is a new thing that was implemented in the wake of the 2008 crisis and quantitative easing.
The federal funds rate is an overnight rate for one. And I'm sure that American banks sometimes use libor for setting rates on securities.
I would be surprised if the Euro didn't have something that was closer to the federal funds rate than libor.
"Eonia reference rates are calculated by the European Central Bank, based on all overnight interbank assets"
GBP: SONIA
I used to work for a small, independent East-European mortgage-broker back when the financial crisis started and pretty much all the banks active on my country's market were using LIBOR as a reference rate when issuing mortgages, this is actually the first time when I hear about EONIA. I remember my boss telling me at some point to write a script that would parse the daily LIBOR rates from the Internet, and me being very surprised that one of few places from where I could get those values reliably was a crappy geocities-like page hidden inside a major bank's website.
First, the Fed doesn't directly control the rates at which banks borrow and lend among themselves. Nominally, the Fed does offer funding to commercial banks, through it's so-called discount window, but it discourages banks from borrowing from it directly by offering those funds at a rate (called the discount rate) that is higher than prevailing market rates. As a result, no one uses the discount window unless they must, which further reinforces banks' avoidance of it—everybody knows that something has gone terribly wrong if a bank must borrow from the discount window.
Ordinarily commercial banks in the United States instead borrow and lend among themselves at a market rate (called the federal funds rate) that the Fed influences by buying and selling treasuries, making less or more cash available for lending in the commercial banking sector.
More precisely, the Fed establishes a target federal funds rate—which is now actually a range, not a single number—and then undertakes asset purchases and sales to coax the actual rate into that target range. But the actual rate at which banks borrow and lends among themselves, which is typically called the effective federal funds rate to distinguish it from the target rate, varies from day to day [0]. The difference between the target and effective rates says something about liquidity in the commercial banking sector as well as something about the effectiveness of the Fed's implementation of its monetary policy.
In Britain, the Bank of England implements its monetary policy somewhat differently, through a scheme that involves targeting a rate it calls Bank Rate (and which others typically call official bank rate or the base rate) and then paying that rate on the reserves of commercial banks that it holds on deposit, or charging them a penalty rate if they deposit too much [1]. Commercial banks manage their day-to-day reserve levels by borrowing and lending among themselves, and that's what sterling LIBOR purports to measure.
Dollar LIBOR purports to measure the rates at which off-shore banks borrow and lend dollars among themselves. Since the substantial dollar holdings of banks outside of US jurisdiction are typically subject to minimal regulation, and in particular are not subject to the Fed's reserve requirements, this rate is somewhat different from the effective federal funds rate.
0. See this page for recent daily effective rates versus the target range: https://apps.newyorkfed.org/markets/autorates/fed%20funds#Ch...
1. See this page for an overview, and the linked "Red Book" for details: http://www.bankofengland.co.uk/markets/Pages/sterlingoperati...