(KCBD) - It's not in any official dictionary or the official NWS glossary, but you have heard a lot about it on TV.
Explosive cyclogenesis is the official term for what is happening. It is defined by rapid or sudden pressure drops similar to what happens in a hurricane during the summer months.
The pressure level must drop 24mb in 24 hours to be classified as "explosive" cyclogenesis or else it is simply a normal or strong developing low pressure system in the Atlantic.
So, in other words, Bombogenesis is not an official term nor is bomb cyclone, but probably came into "popular" existence because of the term "explosive" which suggests an explosion or bomb occurrence took place.
In this case, explosive cyclogenesis is simply a storm system that is rapidly or suddenly strengthening much like a hurricane does.
There is a brief explanation on the NOAA Ocean Services page referring bombogenesis as a "popular" term but not an official term.