Bullying happens in school, the work place, home, kids etc.. is there a mental health diagnosis for why people bully others?

Nonspecific: Bullying can be a final common pathway of behavior for a wide variety of reasons. There is no specific psychiatric diagnosis that matches bullying. Certainly things in such a person's history -- like being mistreated or abused at home -- may be a factor. But it's not an excuse, and bullying must be stopped. If the bully has been through trauma, s/he needs help as well as firm limit-setting.

Inferiority: Believe it or not, bullies tend to have inferiority complexes, often from having been bullied themselves. and yes, it can continue into adult life. Tell someone you know and trust about the bullying. Keeping it inside does no good. There is typically no mental health diagnosis, per se.