Cord Blood is saved to preserve stem cells. These cells can be reproduced and have been found VERY helpful in curing diseases in children like luekemia. Because the cells are banked and come directly from the person who going to use them, they are a perfect match. There is a 50/50 chance that other family members may also be able to use the stem cells. Scientists say that the future uses of Cord Blood will greatly increase with time. The downside- Banking Cord Blood is expensive. I think we paid around $2000 to bank it with a reputable company, and then pay $110 anually to keep it banked. It can be saved for up to 18 years, and maybe longer. Some research shows that these kind of cells will be helpful in curing some neurological problems like Alzheimer's, but I don't know of anything like that yet.
Hope this helps!