Adult stem cells are obtained from tissues that have already undergone some degree of development. Despite the name, they come not only from adults, but from fetuses, newborns, and children as well.
One rich source of adult stem cells is the umbilical cord, an organ that naturally dies after birth. Removing stem cells from an umbilical cord after a baby has been born is therefore not controversial, as long as the parents consent to the procedure.
Another source is bone marrow. Several types of multipotent stem cells are found there. Obtaining bone marrow from a donor is a painful procedure that has some inherent risks, but if the donor (or the donor's legal guardian) gives informed consent, ethicists generally consider it acceptable.
As long as informed consent is obtained from a person (or from the person's legal guardian), the use of adult stem cells is uncontroversial because they can be obtained without killing an embryo, child, or adult.