Those of you with authoritarian personalities are invited to leave the way you came in.
Demarchy is a form of decentralized participatory self-government first proposed by John Burnheim in his 1985 book Is Democracy Possible? The Alternative to Electoral Politics. It was conceived as an alternative to and remedy for the anti-democratic tendencies of large-scale representative "democracy" prevalent today.
A community or populace implementing demarchy will first dispense with the idea of a head of state . A demarchic governing body is composed of councils, with one council for each facet of community life for which governance seems necessary. There may be councils for education, public works, trade and/or other economic issues, etc. The members of these councils are not elected; rather, they are chosen for a short term by the public at random from a group of volunteers* (not unlike a jury of the willing).
Herein lies demarchy's major departure from most modern forms of representative democracy. Burnheim's view is that the majority of the problems with representative democracy as it is are caused by the very fact of elections, re-elections, incumbency, career politicians and, in fact, voting itself. In short, politics is detrimental to governance.
Governments composed of those who seek power are more likely to serve the powerful than the general public. Communities are ill-served by their decision-makers being geographically and socially removed from the community. What works in municipality A may be detrimental to hamlet B. Demarchy makes an attempt to eliminate these problems by placing governmental power directly into the hands of the citizenry while avoiding the potential impracticality of pure direct democracy.
As mentioned, the council members are randomly selected from a group of volunteers. This drastically reduces the capabilty of "career politicians" to wield power for any sustained amount of time, as councils would be re-staffed at regular intervals, and diminishes the possibility of powerful groups or individuals "buying" an office. That council members are volunteers is important; it ensures that those who don't want to govern will not have to, and selects for knowledgeable people being on the appropriate councils (that is, folks who are interested in education would volunteer for the education council, etc.).
Since there is no head of state or other supreme executive, the councils have jurisdiction over and implementory responsibility for their area of influence. Councils would have some sort of networking structure for communicating, coordinating and cooperating with each other. Decision making, "legislation", rests with the councils. Items could be brought up for public vote if the community chooses to implement demarchy that way, but popular vote is not an inherent part of demarchy. Burnheim believed voting to be at best nominal assent of pre-determined outcomes and at worst sham participation.
Demarchy is a rough outline for a participatory democracy; details would necessarily be fleshed out to serve the needs of the community involved. Term limits, selection methods to ensure proportional demographic representation, community standards, methods of council recall, and many other things would have to be decided and/or remain amendable throughout.
The primary benefit to demarchy is the necessity and encouragment of actual participation and the reality of citizen empowerment. Given a community of appropriate size, it is probable that everyone would hold a council position at one time or another. However, demarchy requires a polity committed to self-governance and vigilant against apathy. As such, it is probably most appropriate for small and/or intentional communities.
*Oolong asked about the nature of "volunteer" and whether or not that was intended to imply that the council members were working for free. The term "volunteer" was meant only to refer to selection of a pool of potential council members (as in, saying "yes, I will do that if selected") and wasn't meant to refer to compensation at all. I suppose the compensation issue would need to be worked out by the demarchy involved.
DejaMorgana suggests that in a small enough group the same few people would continue to volunteer, causing a lot of the same political problems demarchy exists to address. I agree that this is certainly a problem that would need to be dealt with in some communities.