The familiar red blood cell, pictured in the lower right, is formally called the erythrocyte. It was once a living cell, but by the time it gets to your bloodstream it's only a "biological box" that can carry large amounts of oxygen. On the lower left are the platelets. When you have a cut, the platelets bind together and form a blood clot that stops the bleeding.
The white blood cells are a collection of five types of cells that patrol the blood stream and tissues of the body looking for bacteria, viruses and parasites. Together they are known as white blood cells or leukocytes. The most common type of leukocyte is the neutrophil and is the body's front line of defense against harmful bacteria. You have probably seen evidence of neutrophils if you've ever had a cut that got infected. Surrounding the infection is usually a fluid called "pus" which mostly contains the remains of dead neutrophils.
Monocytes will patrol the bloodstream for a little while, but soon develop into macrophages that can actually "eat" bacteria in the body that isn't supposed to be there. Because of this, macrophages are large and able to engulf the invader.
Eosinophils are especially valuable as fighters of parasites. Because of this association, doctors may suspect a parasitic infection if you have a blood test that shows a higher number of eosinophils in the blood than normal.
Basophils are the least common white blood cell. In addition to fighting bacteria, they are involved in releasing histamine, a biochemical that ultimately leads to an increase in swelling. While histamine can have some unpleasant side effects, it's a necessary part of the immune response.
Not pictured are lymphocytes, which are the second most common type of leukocytes. Lymphocytes are commonly found in the blood, but also in the lymphatic system. They develop into either B-lymphocytes or T-lymphocytes and have many functions in fighting bacteria and viral infections.