What is legitimacy of authority?
We obey people further up a social hierarchy
Most societies are structured hierarchically. People in certain positions hold authority over the rest of us, e.g. parents, teachers, police officers, nightclub bouncers.
Authorities have legitimacy through society's agreement.
The power that authorities wield is legitimate because it is agreed by society. Most of us accept that authority figures should exercise social power over others to allow society to function smoothly.
We hand control over to authority figures.
People with legitimate authority have the power to punish others. We give up some independence to people we trust to exercise authority properly. We learn to accept authority during childhood (parents, teachers, etc.).
Leaders use legitimate powers for destructive purposes.
History has shown that some leaders (e.g. Hitler, Stalin, POI Pot) use legitimate authority destructively, ordering people to behave in cruel and dangerous ways.