A "dirty bomb" is an explosive radiation dispersal device that uses a conventional weapon (e.g. dynamite) as its means of dispersing radioactive material. It is not the same as an atomic or nuclear bomb.
Where a nuclear weapon uses fission to provoke an enormous explosion of radiation, a dirty bomb simply scatters radioactive material that is likely to contaminate an area but not cause mass casualties from fire or radiation poisoning.
Dirty bombs are generally referred to as weapons of mass "disruption" as opposed to weapons of mass "destruction" because of their capacity to provoke widespread panic and fear of radiation, as opposed to large-scale casualties.
The local contamination impact of a dirty bomb depends on 1) the size of the bomb, 2) the type and amount of radioactive material, 3) the type and size of conventional weapon used, and 4) the weather conditions at the time the device is detonated and the physical surroundings.
Immediate deaths and injury are more likely to stem from the explosion than from any radiation. However, even low-level radioactive contamination of a city building or few blocks could take months to clean up. In certain cases, buildings may be more costly or difficult to decontaminate than to tear down and rebuild.
For more information on dirty bombs and their potential threat, go here: United States Department of Homeland Security
and here: U.S. Nuclear Regulatory Commission