Depends. Are you talking about private messages or posts in forums you have subscribed to? I note from your profile that you are receiving email notification. That means that you receive an email every time somebody posts in a thread that you have already posted in. If you no longer want to get these notifications, you have to turn them off, via your User CP and also unsubscribe to each thread individually. If you unsubscribe to threads, they will disappear, but you will also no longer receive notification that a post has been made to them.
Private messages can be individually deleted or deleted more than one at a time via the Private Messaging System.