A stative verb is one that describes a state of being, in contrast to a dynamic verb which describes an action. The difference can be categorized by saying that stative verbs are static or unchanging throughout their entire duration, whereas dynamic verbs describe a process that changes over time. Many languages distinguish between these two types in terms of how they can be used grammatically.
The same verb may act as stative or dynamic. An English phrase like "he plays the piano" may be either stative or dynamic, according to context.
Some languages use the same verbs for dynamic and stative situations, while other use different (but often etymologically related) verbs with some kind of qualifiers to distinguish between the usages. A stative verb is often intransitive, while a corresponding one would be transitive. Compare, for example, modern English with modern Swedish and German.