Authority is distributed when more than one or just a few people can make decisions.
The old way: one person makes all the decisions
In traditional organizations, decisions are made at the top of the pyramid and pushed down the various layers of hierarchy.
Often, such decisions are made with a limited understanding of how they will actually impact the day-to-day work of others, based on a number of potentially misguided assumptions.
The best way: people closest to the action make the call
In more modern, flat organizations, people recognize that the group is more knowledgeable than any single person.
That doesn't mean that there is no hierarchy whatsoever or that every decision must be the result of a democratic vote.
Not at all.
What it does mean is that people can make decisions that concern them without having to ask for permission.
And, when there is a decision that needs to be made that might concern others, then there is a clear and transparent decision-making process that makes it easy to collect feedback from others and reach a conclusion together.
The bottom line is: when the authority is distributed, all the decisions of an organization are spread throughout that organization.