to back off is to stop saying something or doing something to someone that might be upsetting them. It can also involve physically distancing yourself from that person. "Back off!" is often yelled in a very angry way, as when someone has reached their emotional breaking point. Synonyms: Give her a break! Just leave her alone! Back down is different. People say, "Don't back down! Tell her exactly what you want!" This is like telling someone to be assertive. It is used in the workplace, for example, to encourage someone to be confident in their ideas.

When a criiminal once tried to mug me ... I told him "back off!" ... then he "backed down" and left me alone.

The meanings of the phrasal verb, "back off" are: 1. to move backwards in order to get further away from something, e.g. Everyone back off and let the doctor through. 2. to stop criticizing someone or telling them what to do, e.g. Back off, will you? I’ll tell you when I’m ready! The meanings of the phrasal verb, "back down" are: 1. to stop asking for something, or to stop saying that you will do something, because a lot of people oppose you, e.g. Neither side is willing to back down. back down on: We will not back down on the decision to strike. 2. remove oneself from an obligation (chicken out; pull out; bow out), e.g. He bowed out when he heard how much work was involved. Sources: - http://www.macmillandictionary.com/dictionary/british/back-off - http://www.macmillandictionary.com/dictionary/british/back-down - Advanced English Dictionary