A pair of compound propositions that always have the same truth value are logically equivalent. Equivalently, we can say that p and q are logically equivalent if "p if and only if q" is a tautology.
Think of the universal quantifier as a conjunction: all values in the universe of discourse must give true evaluations for the quantifier to give a true evaluation.
The existential quantifier is a disjunction: there must only be one of its values in the universe of discourse resulting in a true evaluation of its predicate in order for the quantifier to have a true evaluation.
It is often helpful, when working with nested quantifiers, as thinking of them as looping through all the values in the universe of discourse.
Chapter 1.2: 6, 18. Chapter 1.3: problems 10, 14, 20.