The term "adoption papers" has different meanings to different parties and at different points in the adoption process.
A birth mother relinquishing her child signs forms known as the Relinquishment of Parental Rights - commonly referred to as adoption papers. These forms can be signed no earlier than 48 hours after the birth. Once signed, they are final. The birth mother can not change her mind after these papers are signed. The birth father's involvement in relinquishment will vary depending on a number of factors, but he has a legal right to be included in the process.
Adoption papers can also refer to the documents collected by those undertaking an adoption, who must gather together birth certificates, police and Social Service clearances, a home study, and other papers as required. Those involved in international adoptions will need to add documents from Immigration and Naturalization to their list of required materials.
But in most cases, when people discuss adoption papers they are referring to the certificate of adoption and any accompanying paperwork. If the adoption was done through an agency, the actual documents will vary depending on a number of circumstances, including where you were adopted from and the year in which you were adopted. Often there will be an intake report that details how you came to the adoption agency, medical records from the time you were under the agency's supervision, and legal paperwork.
Agencies are required to keep a permanent copy of all paperwork they might have, and typically the adoptive parents receive a copy of all adoption papers. Once they reach the age of 18, adoptees are legally entitled to see all adoption paperwork the agency might have.
In an independent adoption, where there is no agency involved, adoption papers would again consist of a certificate of adoption and whatever else might be required by the state. Independent adoptions are legal in most but not all states.