The essence of that process is consensus. This is why as many posts have claimed, small groups can be more anarchistic that larger entities. Consensus in an organic manner means that we work with those whom we have consensus, and we work to bring those outside the consensus into the consensus.
The second major idea behind anarchy is its voluntary nature. While someone mentioned the connection to libertarianism, there are differences. Most libertarians speak of the individual as being superior to the collective, and this agrees with anarchy, but while libertarians seldom see advantages to organizations of any kind, the anarchist seeks out cooperative efforts that are voluntary in nature and horizontal in organization. This voluntary nature requires a self-interested (as opposed to selfish) and self-reliant individual, who recognizes the opportunity that cooperation provides for projects beyond what an individual can accomplish.
This is the third, and most difficult, premise of anarchy: the horizontal nature of the collective organization. Horizontal refers to the levels of authority, or the ideal that there are no levels of authority. How is this possible? I would equate it to what I knew in the Marines, that authority had to be limited and rotational. How does that come from the Marines or any military unit? In the military, the commanding officer cannot be expected to be in command constantly, so when he needs to rest, he assigns an officer of the day. The officer of the day has limited but still official command of the unit for a limited period of time. During that time, he or she makes any decisions about the unit. The officer-of-the-day duty rotated among all the junior officers of the unit. This gave them experience in dealing with the responsibility on a limited and rotational basis. In a horizontal organizing process, any position of leadership would be rotated among all the voluntary members of the collective, each having a specific, but limited authority to act and direct the actions of the organization.
How the authority would be limited (both in time and authority) would be determined by consensus, but the idea would be that every volunteer would have the responsibility and opportunity to be in the position of authority.
If this cooperative and anarchistic organization were highly specialized or technically detailed, then volunteers would have a process to demonstrate their expertise before they would be put into the rotation. Again like the military, the volunteer who had the responsibility could be accompanied by an apprentice (someone preparing to be ready for such a rotation) to provide on-the-job training.
Lastly, each of these point to the uniqueness of every organization based on anarchistic and cooperative processes, that the process (organic, consensus-based, and horizontal) provides the means to establish an organization that is horizontal, voluntary and cooperative. No two anarchistic organizations would have to be alike except in these processes.
I don't consider anarchism to be an ideology, because its very name indicates a rejection of ideology, but it is a process that has some very basic premises that individuals must voluntarily practice in order to remain horizontal and organic without a usurpation of the organization by selfish individuals who seek power.
Most of the labor movement begins in very anarchistic processes, but at some point, individuals usurp the authority from the voluntary process for their own selfish ends.