There are two ways you can connect with someone on Facebook: by becoming friends with them, and by following them. 

Being friends means that you and the other person have approved the connection and likely know each other, while following an account means that the person hasn't added you back as a friend.

When someone follows you, they'll only see posts you share publicly. You can turn off the ability for people to follow your account at any time from the "Followers" section of your account settings.