The conditional probability P(x|y) is the probability of some event x given the occurrence of some other event y.
The marginal probability P(x) is the unconditional probability of some event x.
Bayes' theorem states that P(x|y)*P(y) = P(y|x)*P(x) = P(x ∩ y).
See the wikipedia article for details.
See also: Sklar's Theorem about copulas, as well as the wikipedia article. A copula combines the marginal, univariate distributions of a multivariate distribution into that multivariate distribution.