This is commonly called a "second-parent adoption" or, if the couple does marry, a stepparent adoption. Where the adopting couple is married, these adoptions are approved pretty readily, because the couple's relationship is legally valid and the child is already in the home and will stay there even if the adoption is denied.
A number of states, however, still frown upon second-parent adoptions by unmarried couples, so a partner who wishes to adopt without marrying should consult a local family law attorney to get an up-to-the-minute evaluation of the law.
This type of adoption can't take place unless one of the following is true:
If the noncustodial parent is the father, a social service agency will determine whether he has abandoned the child or whether his consent is needed before a second-parent adoption can take place. A father who signs a paternity statement, provides support (if he can), and maintains a relationship with his child is not in danger of the child's being adopted by someone else without the father's consent.
If the noncustodial parent is the mother, the social service agency will have to obtain her consent or recommend that her parental rights be terminated. Unmarried mothers without custody must pay support if they can and visit the child -- or risk losing the child to a second-parent adoption.