If you are listing this person on your return ONLY for the purposes of Earned Income Credit (EIC), but you are not eligible to claim them as a dependent, you would check this box.
John and Mary have a son. John and Mary do not live together. Their son lives with John for the entire year. By court order, as a part of their divorce agreement, Mary and John alternate years claiming their son as a dependent on their returns.
It is Mary's year to claim their son as her dependent. However, because he lives with John, their son will be John's qualifying child for the purposes of receiving Earned Income Credit (EIC).
John will claim their son on his return for EIC, and Mary will claim their son as her dependent. On John's return, he would list his son's information and check the box indicating "This qualifying child is NOT YOUR DEPENDENT."