Joining marginal distributions into a joint one

What are copulas, and how are they applied in quantitative finance?

A copula is a joint distribution on $[0,1]^n$ with uniform marginals; by Sklar's theorem any joint distribution splits into its marginals plus a copula describing dependence. This lets you model each variable's distribution separately and then choose the dependence structure (Gaussian, $t$, Clayton, Gumbel). It is used for default correlation in CDO pricing, multi-asset options and risk aggregation; the Gaussian copula's weak tail dependence was a notable failing in the 2008 crisis.