Is a phrase referring to the basic idea that the largest and most interconnected businesses are so important that a government cannot allow them to declare bankruptcy, because said failure would have a disastrous effect on the overall economy.
Simply put, this idea breeds reckless behavior in any institution which believes the government will intervene (e.g. by bailing out the company) in the event the institution is going to fail. The phrase has also been more broadly applied to refer to a government's policy to bail out any corporation it deems necessary. The term is back to center stage since the start of the recent financial meltdown. One of the biggest US companies referred to as too big to fail is American International Group (AIG).
In a free society that is supposed to embrace property rights and capitalism, having companies that can privatize their profits but socialize their losses is unacceptable. Nothing is too big to fail. Companies wouldn't become the disasters they do if they really had to fend for themselves. However, if the government is going to bail them out then why not be risky and reckless. Are you too big to fail? I'm guessing not.