I recommend this paper as a must read: Li, David X. "On default correlation: A copula function approach." The Journal of Fixed Income 9.4 (2000): 43-54. Here's the PDF. It explains what copula is and how it can be used in the financial application. It's a nice easy read.
This should be followed by an article By Felix Salmon "Recipe for Disaster: The Formula That Killed Wall Street". Here how it starts:
A year ago, it was hardly unthinkable that a math wizard like David X.
Li might someday earn a Nobel Prize. After all, financial
economists—even Wall Street quants—have received the Nobel in
economics before, and Li's work on measuring risk has had more impact,
more quickly, than previous Nobel Prize-winning contributions to the
field. Today, though, as dazed bankers, politicians, regulators, and
investors survey the wreckage of the biggest financial meltdown since
the Great Depression, Li is probably thankful he still has a job in
finance at all. Not that his achievement should be dismissed. He took
a notoriously tough nut—determining correlation, or how seemingly
disparate events are related—and cracked it wide open with a simple
and elegant mathematical formula, one that would become ubiquitous in
Copulas are used to recover the joint probability function when only marginals are observed or available. One problem is that the joint probability may not be static, which seems to be the case with their use in default risk estimation. These two readings demonstrate that. Copulas worked fine in insurance, where the joint is very stable, such as death rate of spouses.