When you 'back out' of a course of action or an ongoing task or project, you leave it or decide not to pursue it any further. This is usually used when the person 'backing out' did so unexpectedly, an action resulting in other people being let down or disappointed.

1. Despite giving me his word that he would come to the party, John backed out at the last minute.

1. We signed a contract, so if you back out of the deal now I will take you to court!
2. Despite our major investor backing out, we believe that the business will be up and running by the end of the year.