A blood type is a description of certain characteristics of blood which depend on certain substances present on the surface of red blood cells. There are 46 known antigens, each of which is described by its own system.
Two important classifications to describe blood types in humans are ABO and Rh factor. Blood transfusions from incompatible groups can cause an immunological "transfusion reaction", resulting in hemolysis, anemia, renal failure, shock, and death.
Overall, the O blood type is the most common blood type in the world, although in some areas, such as Norway, the A group dominates. The A antigen is overall more common than the B antigen. Since the AB blood type requires the presence of both A and B antigens, the AB blood type is the rarest of the ABO blood types. There are known racial and geographic distributions of the ABO blood types  (http://anthro.palomar.edu/vary/vary_3.htm).
The precise reason why people are born with antibodies against an antigen they have never been exposed to is unknown. It is believed that some bacterial antigens are similar enough to the A and B glycoproteins, and that antibodies created against the bacteria will react to ABO-incompatible blood cells.
Another characteristic of blood is Rhesus factor or Rh factor. It is named after the Rhesus Monkey where the factor was first identified in 1940. Someone either has or does not have the Rh factor on the surface of their red blood cells. This is indicated as + or -. This is often combined with the ABO type. Type O+ blood is most common, though in some areas type A prevails, and there are other areas in which as many as 80 percent of the people are type B.
Matching the Rhesus factor in the ABO system is very important, as mismatching (i.e. an Rh positive donor to an Rh negative recipient) will cause hemolysis.
Other important considerations for Rh blood types include situations where a pregnant woman and her fetus are of opposing Rhesus types (one positive, the other negative).
The rare individuals with Bombay phenotype do not express H substance on their red blood cells and therefore do not bind A or B antigens. Instead, they produce antibodies to H substance (which is present on all red cells except those of hh phenotype) as well as to both A and B antigens and are therefore compatible only with other hh donors.
Individuals with Bombay phenotype blood groups can only be transfused with blood from other Bombay phenotype individuals. Given that this condition is very rare to begin with, any person with this blood group, who needs an urgent blood transfusion, may be simply out of luck, as it would be quite unlikely that any blood bank would have any in stock.
A allele gives type A, B gives type B, and i gives type O. A and B are dominant over i, so ii people have type O, AA or Ai have A, BB or Bi have type B. AB people have both phenotypes because A and B express a special dominance relationship: codominance. Thus, it is usually impossible for a type AB parent to have a type O child.
When a type AB parent has a type O child, or when one type A and one type O parent produce a type AB child, it is sometimes mistakenly assumed that the child MUST be illegitimate.
Inheritance in the Bombay phenotype
Another possible explanation is that the child or parent who tests as type O has the very rare Bombay phenotype: they have inherited two recessive alleles of the H gene, (their blood group is Oh and their genotype is "hh"), and so do not produce the "H" protein that is the precursor to the "A" and "B" antigens. It then no longer matters whether the A or B enzymes are present or not, as no A or B antigen can be produced since the precursor antigen is not present.
Rh is inherited the same way, except that it has two alleles and Rh is dominant. Rh Disease occurs when an Rh negative mother who has already had an Rh positive child bears another Rh positive child. The antibodies in the mother's blood react to the infant's blood. This reaction doesn't always occur and is less likely to occur if the child carries either the A or B antigen and the mother does not. In the past, Rh incompatibility could result in stillbirth or death of the mother. Rh incompatibility was until recently the most common cause of long term disability in the United States. At first, this was treated by transfusing the blood of infants who survived. At present, this affliction can be treated with certain anti-Rh(+) antisera, the most common of which is Rhogam.
Blood donors and blood recipients must have compatible blood types. O- is the universally compatible blood type. The chart below illustrates how people with different blood types can receive or donate other blood (X means compatible). An A- person, for example, can receive either O- or A-, and can donate to people with AB+, AB-, A+ or A- blood.
Blood types are not evenly distributed throughout the human population. O+ is the most common, AB- is the rarest. There are also variations in blood-type distribution within human subpopulations.
Other blood types
Other blood type systems exist to describe the presence or absence of other antigens. Diego positive blood is found only among East Asians and Native Americans. MNS systems gives blood types of M, N, and MN. It has use in tests of maternity or paternity. Duffy negative blood gives partial immunity to malaria. The Lutheran system describes a set of 21 antigens. Other systems include Colton, Hh or Bombay, Kell, Kidd, Lewis, Landsteiner-Wiener, P, Yt or Cartwright, XG, Scianna, Dombrock, Chido/Rodgers, Kx, Gerbich, Cromer, Knops, Indian, Ok, Raph, and JMH.
Social significance of blood types
In Japan, some people believe that personality is related to blood type. From the preponderance of some blood type in a population, "experts" claim to be able to deduce the character of that population. The "experts" also believe that they can calculate how well the blood types of different people match. A Japanese employer could therefore aim to get a proper mix of blood types among their personnel.