When we talk about someone's blood type, we're talking about what kind of antigens they have present on their red blood cell's surface.  Often when we speak of antigens we're talking about something foreign, but in this case the antigen is self.  Blood typing is important in preventing transfusion reactions.

ABO Blood Groups

Type A Has A antigen on the RBC surface, and Anti-B antibody in the plasma.
Type B Has B antigen on the RBC surface, and Anti-A antibody in the plasma.
Type AB Has both A and B antigens on the RBC surface, and neither of the antibodies in the plasma.  Because of the lack of antibodies in the plasma, type AB persons can recieve any type of blood as long as the Rh factor is matched.
Type O Has no antigen on the RBC surface, and both Anti-A and Anti-B in the plasma.  Because the cells do not have an antigen, type O blood can be given to anyone as long as the Rh factor is matched.

The Rh Blood Group

Another important antigen that we need to identify is the Rh group.  The blood of an Rh positive person has the antigen on the red blood cell surface, while the blood of an Rh- negative person does not.  Unlike the ABO blood groups, there is no antibody present in the plasma normally.  An antibody will be formed only if an Rh- negative person is exposed to Rh+ blood.

A good example of this is what happens with hemolytic disease of the newborn.  This is only a concern for an Rh- mom carrying an Rh+ baby.  If mom's and baby's blood mixes at birth, mom will form antibodies against the Rh antigen.  This becomes a problem for the next Rh+ baby.  Mom's antibodies will attack the second Rh+ baby's red blood cells causing agglutination.  This is a life threatening situation for the newborn.  To prevent this from happening, mom will be given a shot (Rhogam) after the birth of her first Rh+ baby, and then again after the births of any other Rh+ babies.