O negative is the universal donor, so obviously whatever the blood type of the recipient, there will be no reaction. Similarly, AB positive is the universal recipient, so again there will be no reaction. After this, it depends on the antigens in the donated blood, and the antibodies in the recipient. (The antibodies in the donated blood are not important because a typical transfusion is only packed red blood cells, with little plasma, and the plasma is what contains the antibodies.)
Everyone of type A has anti-B antibodies in their blood, and everyone of type B contains anti-A antibodies. Finally, people who are type O have both anti-A and anti-B antibodies. As a result of this, aside from the two examples above, no one can receive group-mismatched blood. If type AB blood is given to someone with type A (or type B) blood, the recipient’s antibodies will react with the B antigen (or A antigen) on the transfused blood, with serious and possibly fatal consequences. If type A blood is given to someone with type B, the anti-A antibodies of the recipient will cause a transfusion reaction, and the same goes for type B blood given to someone with type A. Since type O individuals have both anti-A and anti-B antibodies in their blood, they can only receive blood from another type O person. Fortunately type O is the first or second most common type of blood in most countries.
At this point many people will be wondering about Bombay blood, despite the fact that it is exceedingly rare. Type O blood does not really lack all ABO antigens. It contains the H antigen, which is modified in type A and type B individuals. People with Bombay blood lack the H antigen as well, so they have anti-A, anti-B AND anti-H antibodies. Their anti-H will react with the H antigen in type O blood, so they can only receive blood from another Bombay blood person.
Things are a little simpler with the Rh part of blood type. Rh negative means they lack the D antigen of the Rh system, and Rh positive means that they have it. Rh positive recipients can receive both Rh positive and Rh negative blood without consequences, for example an A positive person can receive A negative blood, B positive can receive B negative, and so on. Rh negative people cannot receive Rh positive blood, except in the most dire of circumstances. This is especially true in women of childbearing age. While the blood of Rh negative people does not automatically contain anti-Rh antibodies, they will almost certainly develop them after an Rh positive transfusion. This in itself is not dangerous, but should they somehow receive Rh positive blood in the future, they will have a severe transfusion reaction at that time.
This is important for Rh negative women of childbearing age because most people (and therefore most men) are Rh positive, so there is a good chance that her baby will be Rh positive and the anti-Rh antibodies she develops can seriously harm the pregnancy.
As a matter of fact, this used to be a significant problem for Rh negative women carrying Rh positive pregnancies. Some of the baby’s blood often transfers to the mother at delivery, and about 16% of the time she would develop anti-Rh antibodies, with serious consequences to the next Rh positive pregnancy. Advances in prenatal care (the development of Rh immune globulin) has reduced the chance of the mother being immunized to the Rh factor from 16% to 0.1%