An intuitive, visual guide to copulas

People seemed to enjoy my intuitive and visual explanation of Markov chain Monte Carlo so I thought it would be fun to do another one, this time focused on copulas. If you ask a statistician what a copula is they might say "a copula is a multivariate distribution $C(U_1, U_2, ...., U_n)$ such that marginalizing gives $U_i \sim \operatorname{\sf Uniform}(0, 1)$". I personally really dislike these math-only explanations that make many concepts appear way more difficult to understand than they actually are and copulas are a great example of that. The name alone always seemed pretty daunting to me. However, they are actually quite simple so we're going to try and demistify them a bit.