You may recently saw a new up gradation of Facebook on your timeline. Today I suddenly saw the upgrade that they added follow button on right side of the timeline just after Add Friend button. Before it was as "Subscribe" button but it changed as "Follow" button. Before it was saw on Twitter.
If you have a twitter account then you must see it. On twitter one can follow another by clicking on this follow button. When you already clicked, you are following the account and button will show "Following".
On Facebook we request a person to be friend by "Add Friend" button. When that friend accept the friend request, both of our timeline will be shared each other. Also some days back Facebook has introduced "subscribed" button for 18+ people. If anyone subscribed then they will get timeline update of that person.
Now Facebook has changed that "Subscribed" button to "Follow" button. If you are anyone's fiend then you are already following him/her. If you just want to follow that person, you just need to click on "Follow" button then it will show as "following" status. You can see the picture here.