What does “Following” mean in Facebook in the own profile page just below “profile”, in a square box Mentioning this headline ?
Are you referring to the box that says "Following" when viewing another person's profile? This means that you're not friends with the person in question, but rather you're following them. You can see their public updates, but they don't see anything of yours.
You can enable or disable followers of your own account. If you enable them and someone friend requests you, they'll automatically follow you. This lets people see your public stuff without your personal details.
So, you likely friend requested this person and they didn't accept it (yet).