Quantifiers are not statements -- they are symbols in a logical language. We can ask whether formulas are open or closed in our language, but it doesn't make sense to say that a symbol of our language is open or closed.
There are two quantifiers: universals quantifiers (∀) and existential quantifiers (∃). We use these to 'quantify over' variables in a formula. For instance, (∀x)P(x) reads 'for all x, P(x) is true', or, 'everything satisfies the predicate P' and (∃x)P(x) reads 'there is some x such that P(x) is true', or, 'there is something which satisfies predicate P'.
Closed formulas are formulas where every variable in the formula occurs also in some quantifier -- both of the examples above are closed. It is called 'closed' because the variables are not open to interpretation -- in each of the formulas, x does not refer to anything specific, because it is `quantified over' in the beginning of the formula.
Open formulas are ones which have variables that do not occur in some quantifier. For example, P(x) is an open formula, and so is (∃x)R(x,y) -- the first because x is not quantified over, and the second because y is not quantified over.
