Libor stands for London interbank offered rate. The interest rate at which banks offer to lend funds (wholesale money) to one another in the international interbank market. 
Libor is a key benchmark rate that reflects how much it costs banks to borrow from each other. It is the reference rate for about $350tn of financial products, ranging from interest rate swaps and corporate loans to credit cards, mortgages and savings accounts.
Small daily moves up and down in the Libor rate can cause derivatives traders to lose or make money. Since most mortgage and credit card rates are not calculated daily, ordinary people are more likely to be affected if there is a concerted effort to push the rate in a particular direction for a long period of time. 
A bank wishing to borrow funds (or take deposits) from another bank will quote the London interbank bid rate (Libid), while the London interbank mean rate (Limean) is the average of the two.
However, Libor is the most frequently quoted. It is fixed on a daily basis by the main UK banks though it changes all the time as banks agree transactions involving large amounts of funds (usually in Eurodollars). As such Libor has become a key reference rate for all kinds of floating rate transactions, including the issue of floating rate notes. 
The Libor is set every day at 11am by a variety of international banks citing wholesale short-term lending rates - and the most common rates are chosen. Precise details relating to the LIBOR setting can be found at http://www.bbalibor.com
Example of Libor in use
In the syndicated loan market credits are priced at LIBOR+ rates. The higher risk of the borrower the higher the rate. For instance, a low risk AAA borrower may pay LIBOR + 50 basis points whereas a higher risk BBB borrower will pay LIBOR + 250 basis points. 
Libor in the news – Libor scandal
In 2012, regulators from around the world were probing alleged manipulation by big US and European banks of the London interbank offered rate and other key benchmark lending rates. By the beginning of 2013 a worldwide investigation stretching from New York to Switzerland to Tokyo had discovered widespread manipulation of benchmark interbank lending rates by traders and brokers. Barclays was the first bank to settle with regulators. In December 2012 the Swiss investment bank, UBS, had agreed to pay a record $1.5bn to US, UK and Swiss authorities to settle allegations of "pervasive" and "epic" efforts to manipulate Libor.