As far as the ABO blood goups goes, it all comes down to one gene. I don't remember the name. This gene makes a protein that adds a certain carbohydrate to a certain protein (forming a glycoprotein) on the surface of red blood cells. That gene comes in three different versions, A, B, and O. The A and B versions add different carbohydrates and the O version is inactive so it doesn't do anything. Now, each person has two copies of the gene, one from Mom and one from Dad. So, the different possible combinations are AA, AO, BB, BO, AB, and OO. A person with AA or AO will have type A blood because their version(s) of the protein can add only the A carbohydrate. Likewise with BB and BO. AB people have one copy of each of the active forms, so their blood cells will be covered with a mixture of both. OO people have only the inactive form of the enzyme, so their blood cells don't have either one. These constitute the different antigens shown in the chart you linked to.
Now, let's get to the antibodies. Antibodies are produced by B cells. I won't go into the genetic recombination events that happen to produce the wide array of antibodies that the B cells make; that's somewhat complicated and not necessary here. Suffice it to say that during the maturation process, the antibody genes in B cells undergo a recombination event to make a unique antibody capable of recognizing a specific antigen. Therefore, a given B cell is capable of making antibodies for only one antigen. Millions of different antibodies are produced in this way. However, to prevent the immune system from attacking the host tissue, any B cell that produces an antibody that can recognize an antigen present in the body dies during the maturation process (usually, anyway. Autoimmune disease is the result when that doesn't happen.)
Now, back to the blood types: A person's immune system will produce antibodies against antigens not present in their body, so a person with type A blood will produce antibodies against the B version of the glycoprotein. Likewise, a person with type B blood will form antibodies against the A version of the glycoprotein. A person with type AB will not make antibodies against either and a person with type O will make antibodies against both.
To answer your question at the end, none of them will have a "stronger" immune system. It's really just makes a difference in blood transfusions.
An excellent summary.
And, for completion, when it does come to transfusions, you consider the four basic ABO phenotypes based on the genetic blood groups:
"A" blood - blood group AA or AO
"B" blood - BB or BO
"AB" blood - AB group
"O" blood - OO group
Because type O blood has no exposure to "self" A or B antigens, it will have antibodies to both of these antigens as they are regarded as "non-self". Thus, they can only receive blood in a transfusion that is purely type O*, based on this grouping, as other ABO blood types will have one, the other, or both, antigens, which will trigger a serious transfusion reaction in these people.
However, because type O blood lacks these antigens themselves, and because blood is transfused as packed cells only, stripped of virtually all platelets, white cells, antibodies and plasma (or as best as possible

), type O (especially rhesus D negative) is the perfect donor blood for any person, as one can be almost totally guaranteed* a lack of major transfusion reaction. They are the "universal donors" - and O-negative blood is a particularly useful blood product to give in emergency bleeding where cross-matching would take time.
Similarly, people with type AB can receive any blood of any ABO type, as they will not have developed antibodies against any of the AB antigens, so will not produce a transfusion reaction towards those antigens*. They are the "universal recipients".
Needless to say, type A people can only receive blood that is type A, as they have antibodies to blood with type B antigens, and vice versa with type B people with type B blood.
The Rhesus D antigen (one of the Rhesus CDE group of blood antigens), out of interest, is the other major blood antigen to consider. Being Rhesus D negative means you will develop antibodies to the Rhesus D antigen on the blood cells, again causing a serious transfusion reaction, meaning you can ONLY receive blood that is Rhesus D negative (blood that does not contain the antigen).
The Rhesus status is of importance during pregnancy, as very often the Rhesus D negative mother of a Rhesus D positive baby (due to the father being Rhesus D positive) can produce anti-D antibodies following sensitisation by means of a small amount of fetal Rhesus D positive gblood reaching the mother's circulation, and these maternal antibodies can pass through the placenta to the baby, causing a condition called haemolytic disease of the newborn. It is preventable if screened early, and can be treated with immunoglobulin acting against the anti-D antibodies.
* I have placed an important asterisk here because, although ABO and Rhesus D are the most important blood antigens as far as grouping are concerned, there are other blood antigens of lesser prominence, including Rhesus C and E, which can also cause uncommon and rarer transfusion reactions, and in a typical "group and save" blood test, all of these will be looked for, and further testing required if necessary, to identify the perfect matched sample for your patient.