In general, Protestant churches have sought to create churches that are organized in accord with principles in scripture. There were some differences, based in part on whether or not the model came exclusively from the Book of Acts and based in part on the interpretive principle used to read scripture. In most instances, this has involved an attempt to adopt a model of church reflected in the New Testament.
Anglican churches maintained the same structure of organization as the Catholic Church (a hierarchy stretching from parish priest to bishop to archbishop), with the (major) difference that the monarch rather than the pope was placed at the head of the Church. This also made the Church of England a national church, a model followed by other Protestant denominations. Lutherans also maintained the episcopal model of church governance in Germany. John Wesley, who intended not to start a new denomination but to revitalize the Church of England, initially did not want his leadership called "bishops." When the Methodists began ordaining their own ministers in North America, thus separating themselves from the Anglican Church, he wanted district leaders to be called "superintendents." Americans, however, quickly reverted to the title of "bishop."
The organization of Reformed and Presbyterian churches derives from Zwingli's principle that what is not expressed in scripture is not allowed (as opposed to Luther's principle that what is not forbidden in scripture is allowed). Zwingli and Calvin were more radical reformers than Luther when it came to church structure. They, and Reformed Christians who followed them, tried to set up their churches on the model they found of the early Church in scripture—described in Acts and the Pauline epistles. In general Reformed churches reject a model of organization based on bishops in favor of a structure based on elders and deacons. They believe that leadership should be shared between clergy and laity, because of the principle of "the priesthood of all believers."
The Congregational Church has its origins in the Puritans who left England for the "New World." These Christians were Calvinist in their theology. (They were called "Puritan" because they wanted to "purify" the Anglican church of Catholic elements in favor of a purer Calvinism.) They believed that each congregation was autonomous, and that there should be no overarching denominational hierarchy.
The word "Presbyterian" first came into use in Scotland in the 17th century. It is derived from the Greek word presbuteros, often translated as "elder." The Presbyterian Church in America was founded by emigrants from Scotland and Northern Ireland. They were originally distinguished from Congregationalists not primarily by theology but by form of government. Presbyterians have a representative form of government. This is based on Calvin's belief that leadership should be placed neither in the hands of any individual (all of whom were, by definition in Calvin's theology, very fallen individuals), nor in the hands of the masses. In Presbyterian churches the offices of elder and deacon are reserved for the laity (not the clergy). Deacons largely take charge of the ministry to the poor and sick. Elders are elected to a consistory (sometimes called "session"), where they, in conjunction with the minister, govern the church.
Presbyterians have a pyramid structure: each individual church has a consistory or session, composed of members of the congregation. Above each congregation is the presbytery, which consists of elders and ministers from each congregation in a designated area. Above the presbytery is the synod, and finally the governing authority is the General Assembly. It is crucial to Presbyterians that at each level both clergy and laity are represented. This derives from the Reformed principle (that they share with Luther) of the priesthood of all believers. There are no intermediaries between individuals and God, each human is equally a sinner, and each saint (saved human) has equal status. Ministers are called to their offices because of special gifts (preaching, for example), but others have gifts equally necessary for the community, and ministers enjoy no special status.
Like the Congregational tradition, Baptists have a congregational form of church polity, where each congregation has freedom to govern itself and where major decisions within a congregation are usually made in accord with a vote by the members of the congregation. Though they avoid terms like "denomination," some Baptist groups, like the Southern Baptist convention, function as a hybrid between pure congregationalism and denominationalism. For example, they elect representatives to state and national conventions. The national convention selects an executive committee to make decisions in the periods when the convention is not in session.
Some churches identify themselves as "non-denominational." Sometimes such churches are truly independent congregations; other times they link themselves loosely with other churches of similar theology, though without a set institutional hierarchy. They identify themselves simply as "Christians" (rather than, for example, Baptists or Lutherans). Such churches tend to be fairly conservative and call themselves biblical rather than creedal. That is, they believe that the Protestant principle of sola scriptura (the ultimate authority of the Bible) is best upheld and honored by not creating competing authorities such as creeds. This is often also undergirded by the belief that the Bible is clear in its basic teachings, and so there is no need of a creed to interpret it.