Modal logic extends ordinary logic to handle claims about necessity and possibility, not just what's true, but what must be true or merely could be true. Possible-worlds semantics is the tool philosophers built to give those claims a rigorous formal footing: it treats a claim as possible if it's true in at least one possible world, some complete, coherent way reality could have gone, and as necessary if it's true in every single possible world, no matter how reality could have turned out differently.
A possible world is a complete, internally coherent alternative way things could have gone
A possible world, in this technical philosophical sense, isn't a claim about some other physical universe literally existing somewhere; it's a complete, internally coherent description of one way the whole of reality could have been, down to every last detail, consistent with the actual laws of logic even where it diverges from how things actually are. "You could have chosen tea instead of coffee this morning" describes a possible world nearly identical to the actual one except for that single choice and its downstream consequences, while "a four-sided triangle exists" doesn't describe any possible world at all, because it's not just false but genuinely, logically incoherent.
This framework gives precise, checkable meaning to "necessary" and "possible"
Once claims are cashed out in terms of possible worlds, modal logic can define necessity and possibility with real formal precision: "necessarily P" means P holds in every possible world, while "possibly P" means P holds in at least some possible world, and the two notions turn out to be mirror images of each other, since something is necessary exactly when its negation is impossible. This formal apparatus lets philosophers reason carefully and rigorously about claims that ordinary logic, built only to handle plain true-or-false statements, simply has no vocabulary for, distinguishing what merely happens to be true from what couldn't possibly have been otherwise.
What we're still unsure about
That possible-worlds semantics gives modal claims a precise, workable formal treatment is well established, extensively developed logical machinery, in continuous productive use since the mid-twentieth century. What's more genuinely a matter of ongoing philosophical dispute is exactly what possible worlds actually are, metaphysically speaking, whether they're genuinely existing concrete alternative universes, as some philosophers have controversially argued, or merely useful abstract fictions or ways of talking that don't commit anyone to their literal existence, and philosophers continue to actively disagree about which interpretation is correct, rather than there being one settled, universally accepted answer to what a possible world actually is.
This sits inside Possible Worlds & Modal Logic, one of seven topics in Metaphysics, one of five domains in Philosophy, one of seventeen subjects the app can quiz you on.