Many adoption attorneys have flat fee services for simple issues like document drafting, review, and filing. For more complex issues like talking to an adoptions agency or going to court, most lawyers charge an hourly rate. Rates will vary depending on the complexity of your issue and where in the country you live, so negotiate a rate up front with your attorney to avoid costly surprises.
This is a legal process involving a court hearing during which a judge issues a decree that permanently ends all legal parental rights of a birth parent to a child. This must occur before a child is considered to be legally free for adoption. Termination of parental rights can be voluntary or involuntary, that is, with or without the birthparents' agreement. In some states, there is a period during which the birthparent may appeal, if rights have been terminated without his or her consent. The length of that period varies from state to state.
One point that Matt mentioned that I found interesting, and certainly gives the service credibility, is that they have lawyers that use their service as well. The lawyer will sell the service to a client for thousands of dollars and then pay RapidAdoption.com to prepare the paperwork for them. There are some that they work with regularly and have an agreement with and some that just purchase the service from their website and give the paperwork to their clients as if they had done it.