Separation occurs when some classes of a categorical outcome can be perfectly distinguished by a linear combination of other variables.
Separation (called by various names: "perfect-s", "complete-s", also "partial-s" or "quasi-s", and strongly related to the Hauck-Donner effect), is when all outcomes with a particular level of a categorical variable are greater (less) than some value C of a linear combination of predictor variables, and all outcomes with the other level are less (greater) than that same value C.
This phenomenon causes the maximum likelihood estimate (MLE) of coefficients in, e.g., logistic regression (and related variants) to diverge. Suppose we are regressing a completely separated dichotomous outcome on a single variable using logistic regression, the maximum likelihood estimate of the coefficient for that variable does not exist. This is because the MLE of that parameter tends towards infinity, and MLEs do not exist for asymptotic results. Separation causes further problems for Wald tests of those parameters.
