When a father consents to the idea of giving up his parental rights to a child, he and the child's mother need to visit a lawyer in order to ensure that all paperwork is correctly handled and that the termination of rights is legally valid. The details of parental rights termination vary, sometimes greatly, by state, and it's absolutely essential that a lawyer assist you during this process.
While the details will be dictated by state law, the general process of terminating parental rights goes as follows.
Having a lawyer is essential in cases where a father gives up his parental rights because most states require that the father provide informed consent in order to give up those rights. If he and the mother have legal representation, steps can be taken to ensure he does not later argue that he was not fully informed. In other words, if he changes his mind and says he did not know what he was doing, there will be clear evidence that he had a lawyer and that the lawyer assisted him in understanding his rights and making an informed decision on how to proceed.