How do you know when you have been banned from black ops?

Depends what you did.

Swearing, offensive emblems ect. get you (First time) a 6 hour ban.

Racist/Sexcist comments get you a 24 hour ban first time

A repeat of any of the above will get you permantly banned (untill Treyarch decide they want to let you on. Usually when the next CoD game comes out)