Adoption as it is known in the west is not permissible in Islam. It is not permissible to adopt a child an treat him or her as your own in the sense that it carries your name and it inherits you and you inherit it.
However, in Islam we have what is known as sponsoring an orphan. This is done by taking care of it, providing it what it needs from shelter, food, clothing, schooling etc.
However, the child carries its own name and is called after his or her father. We also have to pay attention to the issue of relationships after the child reaches the age of puberty as it is a stranger to the members of the family unless it suckled from one of the members when it was in the age of suckling.