Definition
An interpretation of a formal system is a mapping from its symbols to objects, relations, and operations in some domain, such that the system's axioms come out true under the mapping. When such an interpretation exists, the domain is called a model of the system. The pq-system has many interpretations: hyphens as integers and p as plus is one; but other mappings can also work, and which is "the" intended interpretation is a meta-question.
Why it matters
How it works
To specify an interpretation, fix a domain of discourse and map each constant, function, and predicate of the formal system to something in the domain. Then evaluate each axiom — if every axiom is true under the mapping, the interpretation is a model. A theorem is true in every model of the axioms (semantic completeness for propositional logic). Some statements are true in some models but not others — these are independent of the axiom set.