What Would Have Happened If AIG Failed?

I've heard the claim AIG is "too big to fail." In the comments at Positive Liberty I noted and asked:

And if [AIG] are too big to fail then they were probably too big to be allowed to exist. I know that’s not a free market answer.

But what does free market economics say about firms who are “too big to fail”?

To which AMW answered:

I like to think that there is just economics, but that’s probably naive.

In other words, economics says, let them fail.

But what would happen if they did? Why do folks say they are "too big to fail"?