Normal modal logics can be given a nice semantics by means of Kripke models, also known as possible worlds semantics.
Probably the most important reason for the popularity of possible-worlds semantics is that common modal axioms correspond exactly to certain algebraic properties of Kripke models in the following sense: an axiom is valid in a model if and only if the alternativeness relation of satisfies some algebraic condition. (In fact, the correspondence holds on a higher abstraction level, the level of frames, consult [vB84] for details.) In particular:
The common normal modal logics can be characterized by appropriate classes of Kripke models. In the following theorem, a Kripke model is said to be reflexive iff its accessibility relation is reflexive, and so on.
The common normal propositional modal logics are conservative extensions of classical logic: if a formula does not contain any occurrence of the modal operator then it is provable in a system mentioned in the previous theorem if and only if it is provable in the propositional calculus.