In the US, the Fed determines the federal funds rate, which is used by banks to lend money to each other.
In the UK, I am assuming the Central Bank has the same role.
So why then is there a need for Libor? Libor is aimed at asking the banks how much it would cost them to borrow from other banks. Why ask them, when these costs are already settled by the Central Bank?
Libor is a term rate (eg 3 month libor is the rate at which banks would lend to each other for 3 months). Fed funds (or Sonia in the UK) is an overnight rate. That's the difference.
The reason banks are asked rather than the rate being observed is that transactions do not take place for all of those maturities, at least not with enough banks to make it accurate at any given time. So the best that can be done is ask them at what cost they can borrow and surely each has a bias to either overstate or understate depending on their market positions. Although in general you would not want to tell the market you think you can borrow at a high rate as that will cause the market to expect a higher rate from you.
Libor is the underlying for trillions of derivatives contract and serves a very important function. Like a lot of things in finance such as MTP, we can criticize it a lot but it may be the best we have at the moment...
