The above write-up, while entirely correct, makes it sound a bit
abstract. The universe of discourse is
everything you're talking about, and which you either never
mention, or don't mention again once it's been established.
For example, if you're talking about natural numbers you could say forAll x, y (x < y or x = y or x > y). There is no need to be more precise and say forAll x, y in N, because N is implicit, being the universe of discourse. But if you're talking about a property that holds for natural numbers but not for reals, it is necessary to qualify what the allowable values of x and y are.
It is not a specifically mathematical notion. If all left-handed people have chintz curtains, and all people with chintz curtains holiday in Majorca, then we can make sweeping statements about the holiday preferences of left-handers without having to burden it with footnotes saying "Of course, it's possible that in the future, genetic engineering will create left-handers who prefer cold climates, in which case the whole argument has had the Richard". The universe of discourse "all people" is "all people of current genetic makeup of Homo sapiens who haven't been kidnapped by aliens etc etc...".
The universe of discourse is the domain in which the quantifiers have their scope. For some universe U, forAll x and forAll x in U mean the same; and likewise with the existential quantifier.