Host vs. Graft/Graft vs. Host
Donated blood stem cell transplants bring with them their own distinctive "self" antigens. The immune system of the patient, who is called a host, senses that these antigens are "unmatched" or "non-self." This prompts the patient's immune cells to attack the donated (transplanted) cells, which are called a graft, and this assault can lead to the patient's rejection of the transplant ("host vs. graft").
More often, though, some mature donor immune cells that are mixed in among the transplanted blood stem cells recognize the antigens on the patient's body cells (host) as "non-self." This causes the transplant to attack the patient's tissues and organs. The result is "graft vs. host" disease, which can be very serious.