If you are considering adopting a child, for whatever reason, and would like to know more about the process, it may be helpful to first of all begin by looking at the various avenues of adoption that are available to you.
1.) Private domestic adoptions: In this type of adoption, you would be adopting a child directly from another household, rather than from a foster care agency, with some sort of charity or not-for-profit organization operating as an intermediary between yourself and the other family. In adoptions like this, the parents of the child ordinarily decide to put it up for adoption before it is born, and may take an active role in the adoption process, such as selecting the new parents. In other cases, however, the parents who have chosen to give their child up for adoption may for whatever reason remain anonymous, preferring to deal with the new parents only through the charity or not-for-profit intermediary.
2.) Foster care adoption: This is the type of adoption people are most familiar with (since it's usually how adoption is represented on television). In this case, the child to be adopted has already been - for one thing, born - and placed in some form of public care. There are variety of reasons for children to be placed in foster care; while in certain cases children may have been taken away from their parents due to maltreatment or some other incapacity to care for their children, this is not necessarily the case - and the reasons for parents to put their children in to foster care may not always differ significantly from those who put them up for the type of domestic adoption described above.
3.) International adoption: This type of adoption involves adopting a child from another (usually, but not necessarily, economically depressed) country. This can be done through either public or private agencies. In the United States, the countries from which the most children are adopted from include China, Guatemala, Russia, Ethiopia, South Korea, Vietnam, Ukraine, Kazakhstan, India, Liberia, Colombia, and the Philippines. Each country has different procedures for international adoption, and the procedures in some countries are much better regulated than others; this last point is especially important, considering that countries with poor regulations regarding international adoption procedures have been victim to child trafficking scandals in the past.
4.) Embryo adoption: In this type of adoption occurs when, following in vitro fertilization, a couple donates remaining embryos to another woman or couple in order to assist her or them with pregnancy and childbirth. In these cases, the child born is considered that of the woman who carried the child, rather than that of the embryo donating couple. Legally, embryo adoption is considered as a transfer of property as opposed to an adoption (although in some cases can be considered as an adoption for tax purposes).
In each type of adoption, there are varying degrees of "open" and "closed" adoption: "open" referring to the involvement or knowledge of the parents who gave their child (or embryo) up for adoption, and "closed" referring to a strict confidentiality, or no knowledge of the original parents whatsoever. It is not, however (as it was suggested above) all or nothing; there are various degrees of involvement or knowledge between the adoptive parents and the genetic parents - this is something you may want to consider beforehand, namely, how much you are going to know about the genetic parents and how involved they are going to be in the adoption process.
Once you have decided which type of adoption is right for you, the next step would be to contact the appropriate agency for more detailed information: in the case of the two forms of domestic adoption, contact or visit either the not-for-profit intermediary (in the case of private adoptions) or the foster care agency (in the case of public foster care adoptions). While it is possible in some cases for genetic parents (with the help of a lawyer) to deal directly with adoptive parents without the help of an intermediary, be aware that this is illegal in certain jurisdictions.
In most cases, in the case of international adoption as well, the first step will be to choose an agency. The adoptive parents will then need to be approved by the country from which the child will be adopted; a dossier will need to be prepared which ordinarily includes a background check, fingerprints for a criminal record check, and a home study review by a social worker. Typically parents will need to visit the country from which they are adopting their child during the process (and some countries even require you to move there permanently, although this is not the norm). Finally, once the child has reached its new home, the parents will have to take the steps to make the child a citizen of their new country.
In the case of embryo adoption, you would again need to be approved by an agency which would enter you into their "pool" of potential adoptive parents. At that point, it would be up to the in vitro couple to select you, and only at this point would the specific arrangements, including the openness or closeness of the adoption, be determined.
This is a very brief overview of the different avenues of adoption and of the different processes which you can expect: but it is not the first or final word on the matter whatsoever, and it will be necessary to seek out more detailed information regarding the individual practices and processes of each agency on your own. Contact agencies directly, and request information: for something as important as your future child, don't rely on a web browser search - get it straight from the source!