In Samsung Galaxy Young Duos, It vibrates even if the vibration mode is off. Why is this happening? How to fix it?
I think you've misunderstood what vibration mode is. The sound control has three settings:-
Sound on: the phone will ring and vibrate normally. You can turn vibration on or off for each notification individually in that app's settings.
Vibrate: the phone will still vibrate, but not make noise. In this state, any notification that would make a sound will vibrate instead. If the notification would have vibrated, it still does.
Silent: notifications and incoming calls will not cause vibration or noise.
Any notifications which have vibration enabled by the app will vibrate in either sound on or vibration mode: only silent mode will disable vibration. Also, if Vibrate when ringing is turned on in Settings → Sound, then incoming calls will make the phone vibrate in sound on mode as well as in vibration mode.
May be a bug or something. Restore factory settings after making a back-up of important files.
Note - Check the settings once again. You might have turned on the vibration.