Technically I suppose every mother is a birth mother, since she gives birth. The term birth mother, however, is reserved for those women, usually unmarried, who relinquish their children to be raised by others. Typically, relinquishment is shortly after birth; some birth mothers don't even get the opportunity to see and hold their babies. In some instances, under difficult personal circumstances or by court order, a woman may relinquish a child past infancy.
The reasons women give up their children are varied. Some women are able to take an altruistic view with respect to their own abilities to care for a child. Other women are pressured or coerced into signing relinquishment papers. The circumstances have varied over the years as society's attitude and approach to out-of-wedlock pregnancy has changed.