The biggest financial scam in the history of the world.

"Explain it like I'm five" explanation:

LIBOR (the London InterBank Offered Rate) is the rate at which banks charge each other to lend to each other. A committee made up of 16 banks (some sources say "up to 18") determines the average rate which the banks (within this committee) have been charging each other, and report that as the LIBOR. These numbers are self-reported and are not necessarily based on real trades. They "throw out" the highest four and lowest four numbers and average the remaining eight to make the LIBOR.
Why do we care about the banks trading? Banks lend to each other "to either make a profit or cover any short-term cash shortfalls on the part of the borrower." The banks who are lending the cash make extra profit. If banks have to raise interest rates for lending funds, it means the lending bank is less confident in the loan. This is meant to reflect "the health of the wider banking sector."
For Europeans, you may be familiar with Euribor. This is the equivalent.
However, this committee has been, evidently, lying about it, in both directions (higher and lower than reality.) Because of the fact that numbers are thrown out, it means that it's possible more than just one or two banks are falsifying their numbers.
Effectively, they've been falsifying their credit scores. They've been doing so to make money. Now, "in many cases", these falsifications were made to push rates down, so consumption would go up. They also did this to mask how much financial stress they were under: if they lowered interest rates, it would make them seem more confident. Again: higher consumption, more profit. However, they pushed them higher on many other occasions. They do this, from the evidence we have, at the request of traders. (Comment with more detail on how they make money off this.)
"Every interest rate in the world is based on LIBOR." Ok, not every single one. But "some mortgages and loans are directly linked to the Libor rate." Most of the remaining loans are at least influenced by it, as it is often used as a benchmark, especially for commercial mortgages and loans.
Individual sources of blame aren't clear yet. The recently-quit chief executive of Barclays Plc "blam[ed] a group of 14 traders out of 2,000" and claimed that he "didn't know about their activities until a week before regulators published their findings." However, these requests and exchanges had been going on for years, so there's a lot of questions around why it took so long to come out. This particular bank (Barclays) seems to have born the brunt of the news flurry: I couldn't find much mention of other banks aside from the Bank of England. I did find a reference to an ongoing US aggregated lawsuit against a dozen (+) banks (including Citigroup and BofA), accusing them of this same behavior (reference). However, as this is currently evolving news, it's possible more banks could come under scrutiny. Several others have been implicated thus far: UBS, RBS and Citibank.
TL;DR: LIBOR is the bank-to-bank lending rate. It's self-reported and easy to fake. It has a wide influence in the lending market. We know for sure that traders within one bank faked and manipulated their LIBOR numbers to maximize profits. They did so at the request of other traders.

Sources: http://www.bbc.co.uk/news/business-18622907
http://www.bbc.co.uk/news/business-18613988
http://www.businessweek.com/news/2012-07-04/diamond-says-barclays-penalized-for-being-first-fined-over-libor
http://buzz.money.cnn.com/2012/07/04/barclays-libor-email/