An adoptee is a person who is being adopted. The opposite party in a adoption is called the adopter. A child is most likely to be an adoptee, whereas an adopter is typically an adult who is agreeing to adopt the child and assume the parenting role.
The term adoptee came into popular use toward the end of the 19th century. The word adopt comes from the Latin term "adoptare," meaning "to choose." More generally, the term adopt has come to mean "taking (a thing) on for oneself." For example, the phrase "we adopted the new department policy" would be a valid use of this definition.