Several ways come to mind.
If it's an application notification, you can float your cursor over it and an X will appear on the right side. You click it and you can choose to hide the friend or the program; and hiding the program for one friend will hide it for all friends.
So, if you hide Farmville updates from Bob Starkey, you hide all Farmville updates.
Or, if you're getting emails, you can click on "Account" in the top right corner. Go to "Account Settings" on that drop down menu. Then on the page that loads, click the tab for "Notifications" and uncheck what you don't want.