If you've got a theory that you're trying to prove, and you uncover a conclusive piece of evidence, then the case is closed. Conclusive means you've got your answer, you've proved your theory, and there can't be any doubt about it.
If want to prove that frogs can speak, and you record a frog reciting "The Midnight Ride of Paul Revere," that would be conclusive evidence for your claim. But if the frog mumbled and might have been just making frog noises, critics would say that your video is not conclusive. You can also use the word conclusive to describe winning something easily or by a large margin, like when the judges handed the tap-dancing bear a conclusive victory, and he moved on triumphantly to the finals.
