What Are The Different Blood Types?
The ABO blood typing system is the most commonly known system for determining blood type. This blood typing system was discovered in 1901 by an Austrian scientist named Karl Landsteiner, and it consists of the four blood types A, B, AB, and O.
How were these four blood types discovered? Karl Landsteiner observed certain chemical molecules called antigens on the surface of red blood cells. An antigen is a foreign surface molecule on a cell. Landsteiner observed two specific antigens which he coined antigen A and antigen B. Those red blood cells with only antigen A came to be known as type A, while those with only antigen B came to be known as type B. If the surface contained both antigens it was identified as type AB blood and if there were no antigens it was called type O.
In response to the antigens on the surface of blood cells, our bodies produce complementary antibodies. Individuals usually develop these antibodies by six months of age in order to protect themselves from foreign antigens. An antibody's function is to recognize and destroy foreign molecules. For example, a type A individual would produce antibodies against the B antigen. Therefore when type B blood is introduced to a type A individual, the antibodies will recognize the B antigen and destroy it. In essence, the type A individual will reject type B blood, and vice versa. A type AB individual has both antigens present in their blood so as a result they do not secrete any antibodies, and can successfully receive type A, type B and type AB blood. Conversely, a type O individual does not contain any antigens in the blood and therefore has antibodies against both antigens A and B. People with type O blood can only receive type O blood.

How Is Blood Type Inherited?
Your blood type is determined purely through genetics. An allele is a form of a gene that is contributed by each of your parents. The gene can be given to the child in the form of a dominant trait or a recessive trait. The A and B alleles are dominant while the O allele is recessive. The combination of the two alleles is called an individuals genotype. For example a possible genotype is AO. The phenotype is the trait that is actually expressed. If a dominant allele is contained in the genotype, it is always expressed. A recessive allele is only expressed when both alleles are recessive. As a result, a child with the alleles AO would have type A blood. In order for a person to have type O blood, the genotype would have to be OO.
Here, even though one of the parents has two O alleles the children are either type A or type B
What Is The RH Factor?
The Rhesus (RH) factor is actually a completely different system of classifying blood. It was also discovered by Karl Landsteiner in 1940. The RH system is a little more complicated in that there are about 45 different Rh antigens on the surface of red blood cells. For blood typing purposes, however, RH+ or RH- is determined based on the presence or absence of the D antigen. If it is present on the surface of the red blood cells, the individual is Rh+. Otherwise the individual is Rh-.
The main concern with the Rh+ and Rh- has to do with fetal complications. If the blood type of a mother and her fetus are different, Rh incompatibility can occur. This usually occurs when the mother is Rh- and the baby is Rh+. What happens is that during childbirth some of the baby's blood can enter the mother's bloodstream. If the baby is Rh+ and the mother is Rh-, she will develop antibodies against the foreign blood that can complicate future pregnancies. The first pregnancy is never at risk since it is the one that introduced the foreign antigen to the mothers system. With each subsequent pregnancy the risk for Rh incompatibility increases. The danger comes when the mother's antibodies pass through the placenta and harm the developing baby. Although Rh incompatibility is relatively rare, it is still the leading cause of fatal blood defects in newborns.
How Is Blood Type Determined?
Now that the inheritance of blood types has been explained, how is one's blood type determined? The solution is actually quite simple. A sample of blood is taken from a person and divided between two microscope slides. On one slide, antigen A is added and on the other antigen B is added. By doing this the antibodies in the blood sample will either react with antigen A or B. By reacting with the antigen the blood shows the antigen to be a foreign substance. If the blood reacts with antigen A it is type B blood. This is because type B blood contains antibodies against antigen A. If the blood reacts with antigen B it is type A blood. If it reacts with both, then it is type O blood, and if it doesn't react at all it is type AB.
What Happens When You Cross Blood Types?
When someone is given a blood transfusion it is of the utmost importance that the proper blood is given to the patient. If a person with Type A blood is accidentally given Type B, the antibodies in the person's blood will attack the foreign blood and try to cleanse them from the system. The results can be fatal. Immediate medical complications include fever, shaking, chills, hive, vomiting and shortness of breath. This is why people are usually given their own blood type in transfusions. The exception is type O blood. Since it does not contain either antigen A or B it can be given to a person without having their antibodies detect it. For this reason, type O blood is known at the universal donor. Type AB blood contains no antibodies, so when blood is given to a type AB person, it is always accepted. For this reason type AB blood is known as the universal receiver.
What Is The Bombay Phenotype?
As with most things in life, there is an exception to the rule of blood types. Sometimes a child can display a phenotype that does not correspond to the phenotypes of the parents. For example, a Type A father and a Type O mother can give birth to a baby with the blood type AB. Logically this seems impossible; how can a mother with type O and father with Type A have a baby that is neither type A nor type O? The answer lies within another antigen. The H antigen is the precursor to the A and B antigens. If the H antigen is inherited as recessive (hh), it will never develop into an A or a B antigen. The result is that the phenotype (the trait that is actually expressed in the person) will appear to be O, even though the genotype (the combination of the two alleles) might be A or B. When this occurs, it is possible for a child to have a phenotype that does not correspond with the parents. [Hence, blood tests for paternity purposes are not as accurate as those performed by testing DNA.]

Statistics
The above pie chart contains a breakdown of blood types in the World.

Possible Blood Types
The following chart displays all possible blood types for a child's father given the mother and child's blood type (excluding the possibility of the Bombay phenotype):

The following chart displays all possible blood types for a child given the mother and father's blood type.

Links
General ABO info
http://daphne.palomar.edu/blood/default.htm
Info on blood type compatibility
http://www.ucs.usl.edu
Britannica article
http://www.britannica.com