Antigens are molecules that bind with antibodies - which is to say that they are things which trigger immune responses. In human blood, antigens are protein markers which are part of red blood cells.

Blood antigens are important because giving a patient blood which contains antigens that their antibodies react to will cause the immune system to attack the donor blood as if it were bacteria or another foreign invader. If you receive a liter of blood that your antibodies react to, your immune system will destroy the foreign blood (so you still need blood) and also your kidneys will start to fail because they're being overwhelmed with a liter of dead blood cells all at once.

Most people are familiar with A/B/O +/-, but the reality is far more complicated. Hundreds of different protein markers have been identified in human blood. Not all of them are clinically relevant in the sense that they can cause rejection, and not all of them have easy identification tests. The more precise and thorough the antigen testing can be, the less likely that a blood transfusion will accidentally kill a patient.