My mother received a double lung transplant. It gave us time with her that we otherwise would not have had. Her donor died tragically and I always hope that the letters we exchanged with the family helped to ease their pain. We never told mom what happened to her donor - it would have bothered her tremendously, I think.
We are so, so grateful to that family and we are all donors (Me, DH, kids, etc.).
To the PP that asked why some folks would not want to donate or only donate certain parts, their are some religions with restrictions regarding the condition of the body at burial.