What are marginals? Marginals are mentioned a lot in copula literature, what does the term really mean?
For example what is the intuitive meaning behind a statement like 

"This function describes the dependence structure separated from the
  marginals."

 A: The term 'marginals' is a loose reference to marginal distributions (as you originally supposed).
That's certainly what the quoted sentence is discussing.
More specifically, without any other adjective, the unqualified term "marginals" would refer to the univariate marginal distributions of each of the variables that are related by a copula.
In the case of copulas, two different multivariate distributions can have the same "dependence structure" as measured by a copula. After a univariate transformation of each of the (univariate) random variables involved in the multivariate distribution to uniformity, the joint distribution that results is called a copula. If you then transform each of the individual random variables (changing the "marginals"), the copula is unchanged, even though the multivariate distribution is changed.
A: Marginals are refering to the probability distributions of individual random variables, typically the cummulative distribution function.
I think it seems counter intuitive at first because in most examples you start by looking at the marginals and not the collection, without which we would not be refering to the distributions of the individuals random variables as marginals.
