There are several types of adoptions, each with their specific requirements, including stepparent adoption. But in stepparent adoptions, many of these barriers are removed. For example, in other adoptions, the court requires home visits, hearings, and long waiting periods, however, for stepparent adoptions, the court removes the home visits and hearings. The law is laxer with stepparent adoptions because the child and parent are (1) related and (2) already know one another.
You still need the consent of the both birth parents (unless the other parents were terminated due to neglect, fitness, abandonment, or failure to pay child support). Usually obtaining parental consent is difficult because the birth parent is affirmatively surrendering all of their parental rights (and parental obligations).
If the another birth parent refuses to surrender their parental rights, it is possible to terminate them, but it is a complicated and drawn out process.
Keep in mind that while same-sex marriage is recognized in all 50 states, same-sex parental adoption is not. Some states permit same-sex couples, even stepparents, to adopt children but others do not. You should speak to a lawyer if you are in a same-sex relationship and wish to adopt.
Do you intend to adopt your stepchild? If you are, you may want to speak to a family law attorney. As illustrated above, there are several issues you will need to consider beforehand. You should speak to a lawyer if the birth parent refuses to consent to the adoption. An attorney can go over your situation and help you determine the best way to adopt your child and the legal implications of such an adoption.