In the congregational model, each local congregation is autonomous and governs itself, without interference from higher church bodies. The local congregation owns its own property, and hires and fires its staff without interference. The most that can happen if it does something that displeases a denominational body is that that body can expel the congregation from the denomination. In this form of church polity, the laity are the ones with the most power.
In the presbyterian model, the governance is placed in the hands of small groups of elders, including clergy and a small body of lay elders in each congregation. Power is often distributed among the different levels, with the decisions of a lower body needing to be ratified by a higher body, and vice versa. The local body of elders call the clergy, but this must usually be ratified by the regional body of elders. Property is usually in the hands of the denomination, although there may be provisions for a church to secede with its property if it votes to do so by a supermajority, with the regional body ratifying it. In this form of church polity, the clergy and lay elders are the ones with the most power.
In the episcopal model, the governance is placed in the hands of bishops, who have authority in a given geographical area. It is the bishop who has the authority to ordain, and to place and to move clergy. The property is usually held by the denomination or by the regional unit of government, with the bishop having a lot of the authority over it. In this form of church polity, the bishops are the ones with the most power.
Many denominations do not have a pure form of church polity, but a mix of two or more of the basic types. In the Episcopal Church, it is sometimes joked that the polity is episcopal as regards the bishops, presbyterian as regards the clergy, and congregational as regards the laity!
There are advantages and disadvantages to all three systems.
In the congregational system, there is a high potential for the involvement of the laity, and decisions are made at the lowest level, so there can be a much greater chance of appropriate response to local situations. On the other hand, when dysfunctional situations arise, there is no denominational official who can step in and intervene and address the dysfunction. It is harder to address both congregations that treat clergy abusively, and clergy who are abusive of their congregations. My own father was very dysfunctional, and was fired by four congregations. It would have been very helpful if someone could have stepped in and made sure that he got the help he needed to address his dysfunctional behavior.
Also, while democracy works great in many situations, it can also lead to a “mob mentality” that inflicts harm on minorities. The Southern Baptist Convention, which is adamantly congregational, regularly votes to demonize lgbt folk.
In the presbyterian system, there is the advantage of governing by the wisdom of those most invested in the church. I have met many committed lay elders in Presbyterian churches, and they are a credit to their denomination. However, the danger can be that the process of governance so consumes the energy of those involved, that there is not enough energy for other aspects of the church’s ministry. In addition, it is more difficult to be flexible on the local level, since unity across the denomination tends to be much more highly prized in this system. And there can be a two-tier system of those with power and those without.
In the episcopal system, there is much more efficiency since one person is making most of the decisions, at least for a given region. And therein lies the rub – if the bishop is a person of great vision and wisdom, she or he can provide great leadership and can lead the church to great things. If not, however, the church suffers from the shortsightedness, mediocrity, or dysfunction of the bishop. I think this system works best in small churches, such as many independent sacramental jurisdictions, where the bishop knows everyone in the church, and when there is an absence of a theology that links the particular denomination with the church as a whole or with infallibility, so that people are free to leave if there is a dysfunctional bishop.
I have seen effective examples of all three types of governance, and disasters in all three. So it is not so much the type of polity that is most important to whether a church is doing its work effectively, but how those who govern use that polity.