LIBOR means the London Inter-Bank Offered Rate. It refers to the interest rate that banks charge when they lend money to other banks, as opposed to outside borrowers. The British Bankers' Association sets the rate, which changes every day. It is calculated using information supplied by several larger banks, which itself is based on market and exchange rate fluctuations around the world.
There have been scandals this year about the way the LIBOR is set. Some banks, including Barclays, were found to have given false information, mainly trying to get the rate set lower than it should be so they could borrow money more cheaply. As many loans such as mortgages are tied to LIBOR, this has had an impact on consumers - although there are so many factors that lead to the way interest is calculated, it is very hard for a consumer with no special financial knowledge to understand if, or how, his or her own loans, investments etc are are affected.