What would happen is someone got a blood transfusion of the wrong type?

Although I am no expert on medical subjects, I know that your body will reject the new blood like a new organ. I'm pretty sure said person would die.

The effect or event is known as known as a hemolytic transfusion reaction.  Antibodies against blood group antigens (there are many more than just the ABO and Rhesus factors), present in the recipient's plasma, would hemolyze the RBC's of the donor.  Literally break down the cells.

What would happen generally as to physical effects would be a general feeling of anxiety or doom, breathing difficulty, flushing, fever/chills, severe pain in the lumbar spine region.  General symptoms of shock would set in (drop in BP, rapid and thready pulse, cold and clammy skin).  Products of blood breakdown, especially free hemoglobin, and bilirubin would be elevated.

The prognosis depends largely on how much of the blood product was given, what kind of incompatibility was present, the physical health of the patient before the event, and supportive treatments.  If the reaction is severe, the prognosis is poor.  Often if there is recovery from anything more than a very mild reaction, there will be a significant and permanent kidney damage, due to the excretion of higher than normal levels of blood products and nitrogenous waste.