To switch up what notifications show up on your phone, head to the settings sections within your Facebook mobile app. Go to Settings > Account Settings > Notifications > Mobile, then go through and turn on or off the ones that do and don't matter to you.
Are notifications interrupting you all damn day in the corner of your computer screen, even when Facebook isn't open? This means you've enabled notifications within whatever browser you're using. Open your Facebook settings in a browser window, and go to Notifications > Desktop and Mobile. Under the section labeled "Desktop," turn them off -- or, if you'd rather they just cool it for a bit (say, you'll be doing a presentation using your computer), you can "Mute" desktop notifications for up to 24 hours.