One parent gets legal custody of the children. The other gets parenting time (visitation) rights. This is the arrangement in most cases.
Both parents have joint custody. With joint custody all or most decision-making about the child is shared. Joint custody does not mean that the child must spend equal or substantial time in each parent’s home; a joint custody order can say that one parent’s home is the child’s primary home and that the other parent gets parenting time. Child support can still be awarded if there is joint custody. In Oregon a court cannot order joint custody unless both parents agree to all the terms.
In families with more than one child, one or more children live with one parent and one or more children live with the other parent. (This is sometimes called split custody.) Judges usually don’t order this kind of custody arrangement. They are worried that it may be harmful to the children to separate them.
Very rarely, a nonparent (or “third party”) can be awarded custody in a divorce, or in a separate lawsuit.