Hair bumps are often referred to as ingrown hairs or razor bumps. The terms are often used interchangeably and while a hair bump can be referred to as a razor bumps both are somewhat different from ingrown hairs defined here.
Wikipedia gives a detailed overview of pseudofolliculitis barbae or hair bumps in this article. Once again let us quickly differentiate between hair bumps and ingrown hair.
Hair bumps or razor bumps occur anywhere on the face or body where shaving or other types of hair removal has occurred. Due to several special hair bumps causes the hair can curl back into the follicle prompting an infection which results in pimple like reactions on the surface of the skin. The reactions or pimples can be pustules (with fluid inside or papules, elevated without the pus).
When the hair grows out of the skin’s surface and grows back into the skin this scenario is referred to as extrafollicular.
In the case of ingrown hair, the shaved hair will not have had an opportunity to break the surface of the skin. Due the its curly nature it would have curled and continued to grow under the skin, again causing infection and the resulting hair bumps. This scenario is referred to as transfollicular.
Here is another brief description of hair bumps at Wikianswers.