There are four main blood groups - A, B, AB, and O - determined by the presence or absence of antigens A and B on red blood cells. The Rh system involves the Rh factor antigen D, with Rh+ blood containing the D antigen and Rh- blood lacking it. Blood typing and cross-matching tests are used to determine compatibility between donor and recipient blood to ensure safe transfusions and prevent complications from incompatible transfusions like hemolysis, jaundice, and fever.