Dirty bomb refers to any nuclear bomb where a significant amount of radioactive waste is generated. Early bombs, due to their ineffeciency, are thus by definition dirty. The definition typically doesn't refer to a specific design, but is instead used to contrast with newer, more effecient, bombs.
Another usage of the term is to talk about radiological weapons, but this is an incorrecy usage.
A dirty bomb is a conventional bomb used to distribute radioactive materials. That is to say, it is not a nuclear weapon and no chain reaction takes place, but the bomb site becomes contaminated and the general fear of radioactivity will cause most of the harm.
The term was put in focus on June 10, 2002, when U.S. officials announced they had captured an al-Qaida terrorist in Chicago a month earlier, who was allegedly planning for such a dirty bomb.