If you're convinced that the moon is made of cheese, then you completely believe the moon is made of cheese. Let's hope you're not easily convinced of such silly things!
When we are convinced of something, we really, sincerely believe it. We think it's true. For example, if someone told you the President used to live underground as a mole person, I hope you wouldn't be convinced too easily. Unfortunately, even when someone has convinced us of something, that doesn't mean it's true. Someone who can be convinced of absolutely anything, no matter how untrue, is called gullible.