I might have been a vicar - that is, although I am not actually a vicar, my being a vicar is possible. The idea of a possible world in something like the contemporary sense is normally credited to Leibniz, who thought that God chose this world, from an infinity of possible worlds, to be the actual world. Since God must choose the best, this world is therefore the best of all possible worlds - the doctrine famously satirized by Voltaire in Candide . Possible worlds became a focus for philosophical interest in this century with the development, by Saul Kripke and others, of a semantic interpretation for modal logic . Intuitively, these formulae should be interpretable as saying something about what is necessarily or possibly true.
Possible worlds provide the answer.
The essential idea is fairly intuitive. Something that is merely possibly true, such as 'I am a vicar', is true in some possible situation. There is no impossibility in the idea of a situation in which I am a vicar. This suggests a way of reducing problematic modal claims into claims that do not contain any modal notions. If we are to reduce modality, we must have an independent account of what these possible worlds are.