FAQs ABOUT BLOOD AND BLOOD PRODUCTS:
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Click if it is also a link, for greater details)

What are the different blood components?
What is a "blood type?"
What types of red blood cells can different types get?
What are the different types of "transfusion reactions?"
What are the infectious disease risks of blood transfusions?

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Risk of infection from a single transfusion for selected diseases
Human Immunodeficiency Virus (HIV) 1:2,000,000
Hepatitis C (HCV) 1:1,600,000
Hepatitis B (HBV) 1:200,000
HTLV I/II 1:650,000
Transfusion reactions occur when the blood recipient has an adverse reaction to the transfused blood product. Reactions may be mild (not even noticed) to severe (life-threatening or fatal).
Type of Reaction What happens How common?
Allergic The plasma contains something the recipient is allergic to (maybe the donor ate strawberries for lunch). Usually just itching and/or hives. Treat with Benadryl. 1:30
Febrile The recipient reacts to white blood cells (or cytokines) in the transfused products. Fever greater than 1.8 F above baseline. Treat with Tylenol. 1:60
Delayed Hemolytic Transfusion Reaction When the recipient makes antibodies against one of the antigens on the transfused red cells (not A or B antigens). The patient's system slowly removes this blood from the body. In the future, the patient will receive red cell products selected so that they lack the inciting antigen.  
Acute Hemolytic Transfusion Reaction This is the bad one. If this happens, it is most likely because someone made a mistake. The recipient receives ABO incompatible red cells, and his antibodies destroy the blood cells within the circulatory system. This releases a lot of chemical mediators that can shut the kidneys down, make the blood pressure drop dangerously low, and even cause death.  
A person's "blood type" refers to the combination of antigens (two types, designated "A" and "B") on his red cells, and the antibodies present in his plasma. Blood is made up of red cells, which have the A and B antigens, and plasma, which has antibodies against the A antigen (anti-A) and against the B antigen (anti-B). A person's plasma contains antibodies for the antigens his red cells lack. Group O designates the absence of both A and B antigens on the red cells.
BLOOD TYPE ANTIGENS ON RED CELLS ANTIBODIES IN PLASMA
O NONE Anti-A, Anti-B
A A Anti-B
B B Anti-A
AB A and B NONE

Blood is composed of:
"plasma"--the liquid portion that carries antibodies and coagulation factors;
"red cells"--that carry oxygen and give blood its color; and
"platelets"--smaller than red cells, and help in clotting.
When you donate one unit of "blood," it gets separated into red cells, plasma and platelets,
and can help three different people.
THE BASIC BLOOD COMPONENTS AND THEIR USES:
RED BLOOD CELLS: Used to increase oxygen-carrying capacity in patients who have lost a lot of blood or have low "hemoglobin" levels due to anemia of other causes. Should not be used for "volume replacement"—solutions like "normal saline" are better for that.
PLATELETS: Used to help clotting in patients with low platelet levels, or dysfunctional platelets.
FRESH FROZEN PLASMA: Replaces coagulation factors (clotting factors) in patients with low levels of many factors. Single replacement factors are available (and a better choice) for most single-factor deficiencies.
CRYOPRECIPITATE: (a derivative of plasma) Used for replacing fibrinogen or von Willebrand's factor.
FOR RED BLOOD CELL TRANSFUSIONS
PATIENT GROUP FIRST CHOICE (IN ORDER OF PREFERENCE) SECOND CHOICE IN EXCEPTIONAL CIRCUMSTANCES CANNOT RECEIVE
O+ O+ O-   A+, A-, B+, B-, AB+, AB-
O- O-   O+ A+, A-, B+, B-, AB+, AB-
A+ A+, A- O+, O-   B+, B-, AB+, AB-
A- A- O- A+, O+ B+, B-, AB+, AB-
B+ B+, B- O+, O-   A+, A-, AB+, AB-
B- B- O- B+, O+ A+, A-, AB+, AB-
AB+ AB+, AB-, B+, B-, A+, A- O+, O-    
AB- AB-, B-, A- O- AB+, B+, A+, O+  

Note that group O people can ONLY receive group O, and that group AB can receive ANY group.
That is why group O is referred to as the "universal donor" for red cells. This is the type of blood that is kept in emergency rooms for times when it is needed immediately, before we have the chance to test the patient and figure out what blood type he or she is.
It would be great if we could just have one big supply of O-negative for everyone, but there just isn't enough of it. One of the challenges of running a transfusion service is being responsible about how resources such as group O red cells are utilized. That is why the group O cells are listed as "second choice" for non-O patients—not because they aren't as good, but because they should be conserved if at all possible.
For compatibility of plasma, click the link.