What reasons is it both legal and appropriate for employers to fire someone?

This depends upon what conditions existed, prior to the firing of an individual. There are certain laws enacted that protect an individual(s). It also depends on what typeof employment contract you (or the person affected) accepted as terms of employment. There are employment contracts out there that can employ and fire at will with no cause. Each situation is different, and since you question is too generalized, this is the basic information I can supply for the time being.