What determines blood type?

Your blood type is determined (genetically) by the presence or absence of specific surface antigens on the membrane of the RBC.
Before a blood transfusion can be administered, it is important to determine if the blood types of the donor and recipient are compatible. If antibodies in the plasma meet their specific antigens on the red blood cells, the blood will agglutinate and hemolyze in a cross-reaction or transfusion reaction. It most likely will be fatal.

The most important RBC surface antigens are A, B and Rh. There are 4 blood types. A, B, AB, and o.
I used a lower case font for O as it is recessive to A,B, and AB. Some times it is expressed as ii.

A type can carry O and will be shown as either AO or Ai, B type can carry O and can be shown as BO or Bi. AB doesn't have a recessive and is only expressed as AB.

Blood types also may have in their plasma (the liquid portion of their blood) antibodies.

ii (O) will have antiA and antiB antibodies. But will have no antigens on their cell membranes.

AB will have no antibodies in their plasma. But will have antigens A and B on the cell membranes.

B will have the B antigen in it's cell membrane and antiA antibodies.

A will have A antigen and antiB in the plasma.

The best way to use blood for transfusion is to use the exact blood type including the Rh factor (sometimes called D).

This can not be done in all situations so O- is used as the universal donor. Notice that d o n o r has two 0's. AB is called the universal recipient.

There is one more thing to understand, since O- has antibodies for A and B, it must be given as packed red cells with as little plasma as possible.

As far as Rh factor, you either have the factor (Rh+) or not (Rh-).