A parent with "sole custody" of a child has exclusive physical and legal custody rights concerning the child. Sole custody arrangements are rare, and are usually limited to situations in which one parent has been deemed unfit or incapable of having any form of responsibility over a child -- for example, due to drug addiction or evidence of child abuse. In sole custody situations, the child's other parent (also known as the "non-custodial" parent) has neither physical nor legal custody rights, but may be entitled to periods of visitation with the child (though those visits may be supervised, especially in situations involving domestic violence or child abuse).
Example: Mother and Father have divorced, due to Father's substance abuse and addiction. Mother seeks and is granted sole custody of Child. This means that Mother alone has legal authority to decide key issues related to Child's upbringing, and Child will live exclusively with Mother. Father may be entitled to visitation with Child.