To add to the existing answer, there's also epistemic logic, where □ is interpreted as knowledge (relative to a given subject).
Possible worlds in such a system are worlds that are consistent with the subject's current information. Since necessity is truth in all possible worlds, it means in epistemic logic that if something is necessary then it just follows from your information (its negation is inconsistent with your information). At least in an idealized sense, necessity is knowledge.
"□p" in epistemic logic means that you know that p, and "□□p" means that you know that you know that p. The two are not the same. For instance, if p is the statement that it's raining, then by knowing p you know something about the weather, but by knowing that you know p you know something about yourself. It's not obvious whether □□p follows from □p. The claim that it does is controversial in epistemology and is known as the KK principle.