Let's look at how you can decide what notifications you want to get in this section:

You’ll recognize this as the place you went to in order to edit your profile. #1 is a list of all the sorts of notifications you can get. #2 and 3 are options of how and if you would like to receive them. For some, you don’t have a choice.  You’ll get a system notification no matter what. System notifications are the ones that appear in your inbox. These include messages from friends/members and friends requests. You can also check the box to be emailed to an outside email address when these come into your inbox so you’ll be sure not to miss them. For the other listed messages you can choose if you’d like to have notification to an outside email address as well. As you can tell, I’m not big on getting emailed. Hit Update Notifications when you're done to make sure to save.