Cord blood is the blood in the umbilical cord and after birth. It is taken after a live healthy birth only, when it is no longer a living part of mom or baby. There is no ethical issues with cord blood because it must be removed. If the umbilical cord is not shut, or the after birth not devlivered, it will actually cause massive problems.
Cord blood has been used to do several different things, But I used cord blood stem cells in my hematopoetic stem cell transplant (hematopoetic means blood... its a fancy term for a bone marrow transplant, but with a slightly broader meaning.) I had no matches in the marrow registry because I have a rare tissue type. Cord blood does not have to be matched as closesly as adult marrow or peripheral blood stem cells (a type of white blood cell in the circulating blood). And since I am an adult, I had to use 2 units of cord blood in my transplant (it is still in trials, I am a part of the clinical trials on this). One birth, one unit, which means I had two different donors for my transplant.
http://www.marrow.org is the National Marrow Donors Program which can give more info on stem cell transplants and the sources of stem cells used