Authority is a formally granted power to make decisions. Authority is generally bestowed upon a position rather than a person. It is different than accountability.
For example, a police officer has legal authority while he holds that job. His authority ends when he retires.
Authority differs from leadership in that the latter is an attribute of the person. A person with strong leadership skills will be able to sway people even without formal authority.
In Lean environments, team members should be granted formal authority to make decisions about their processes. The goal is to get the team members to make decisions on their own. The more skill they exhibit, and the more they get a track record of making good decisions, the more authority they should receive.