You get to decide which stories appear on your timeline. To hide a story so that it no longer appears on your timeline:

1. Hover over the story and click
2. Click Hide from Timeline

Note that stories you hide will still show up in your activity log, which only you can see, and still be eligible to appear in your friends' News Feeds. If you would like to remove a story you posted from Facebook altogether, you can do so by selecting Delete Post.... If you believe that something someone else has posted violates the Facebook Terms, you can report the story.