I generally only do step-parent adoptions. Those take place when a parent marries another person and the new spouse wishes to adopt the children of the previous relationship. The other biological parent can consent to terminate his or her parental rights, and in so doing, has no right to have a continued relationship with his or her children, but also has no responsibilities, e.g., child support.
When children are younger, and child support is a burden for the consenting parent, sometimes this can be a good alternative for the children and the parties. The consent statute associated with step-parent adoptions is found at UCA Section 78B-6-120.