Context: statistics, probability, dependence
To make things simpler, we shall first start off with two random variables. Later we can generalise this to n random variables.
Suppose you have two random variables, X and Y, which has the distribution function, F(x) and G(y), and has a joint distribution function H(x, y). Now by Sklar's Theorem, there exists a function C(a, b) (which satisfies the property of being a joint distribution function for two separate) random variables taking values in [0, 1]) such that
H(x, y) = C(F(x), G(y))
and the function C is called a copula. Its use is to couple two marginal distributions together to form a joint distribution.
Of course, when you replace the random variables X and Y with a random vector x, you get a n-dimensional form of Sklar's Theorem.
There is a more formal form of the definition of what a copula is, but the way described here should be sufficient for most practitioners.
Copulas are especially useful in studying dependence between random variables, something in which statisticians are always interested.