Universal and Existential Quantifiers of First-Order Logic

That sentence is full of jargon. You can find a description of universal and existential logical quantifiers here.

1. A Universal Quantifier is a logical statement that applies to all elements of a set.
2. An Existential Quantifier is a logical statement that applies to at least one element of a set.

You can also look here for a quick description of first-order logic. The term is meant to separate first-order from higher-order logic:

1. First-order logical statements are the usual ones; they act on members of a set.
2. Higher-order logical statements act on other logical statements; think of them as meta-logic.