LIBOR, or the London Interbank Offered Rate, is a benchmark interest rate intended to represent the cost of short-term lending between banks. That is, the LIBOR rate is what banks would charge each other to borrow money - presumably, a lower rate than what they would charge companies or individuals. LIBOR is the most widely used benchmark rate in the world - over $360 trillion of financial products are tied to it.
LIBOR helps determine what the interest rate for most financial products will be. Typically, the interest rate is set as LIBOR + a spread of basis points.* This allows financial products to vary in cost with economic conditions, but also means that the LIBOR rate's fluctuations can cost a lot. For example, a loan might have the interest rate be set as LIBOR + 200 bps (basis points). If LIBOR is somehow at 0%, then the interest rate will be 2%. But if LIBOR is 3%, then suddenly the borrower is paying 5%. On a $200 million or billion dollar loan, this can turn out to be a huge amount of money.
LIBOR is set daily by a myriad of banks communicating their cost of borrowing for over 150 different categories. Usually just a few people are responsible for communicating these rates, leaving them very vulnerable to manipulation. In fact, they have been consistently manipulated! The LIBOR scandal uncovered that bankers (and thus banks) had been submitting erroneous rates in exchange for favors or meals. Big deal, you say, but "a study from Mark Schweitzer and Guhan Venkatu at the Cleveland Fed looked at survey data in Ohio and found that by 2008, almost 60 percent of prime adjustable rate mortgages, and nearly 100 percent of subprime ones, were indexed to LIBOR." Meaning the cost of getting a mortgage will rise or fall for regular people based on LIBOR rate inputs. Here's a great graphic about the LIBOR scandal, and another simple article.
*There are 100 basis points to 1% - a basis point is .01%.