To define what it means to be Presbyterian, we start first by proclaiming - Jesus Christ is Head of the Church, and the Bible is the unique and authoritative witness to Him.
Deeply influenced by Reformer John Calvin (16th c., Geneva, Switzerland), Presbyterian as a Protestant Christian denomination takes its name from its method of governance. It is ruled by elders. Presbyterians do not have bishops (episkopos in Greek), but rather elder leaders (presbyteros in Greek, presbyter in Latin) who are ordained to functions. Ruling elders are ordained to govern. Teaching elders (credentialed with a Masters of Divinity that includes the study of Hebrew and Greek, the call of a church, and the vote of a Presbytery) are ordained to teach, preach, and administer the Sacraments.
There are only two Sacraments held in the Presbyterian Church: Baptism and the Lord's Supper. that arises out of Calvin's interpretation of Scripture as he identified Jesus Christ as Head of the Church, and set Jesus' life as the pattern for church life. Jesus was baptized, and Jesus hosted his Last (or Lord's) Supper. The church is the gathering of baptized disciples, following Jesus into God's future. How? By prayer and decisions made by the elders currently serving on a local church Session, which is moderated by a teaching elder. The elders seek to discern the will of God in Christ in order to journey forward.
At every governing body, or Council, beyond the local church Session, teaching elders and ruling elders vote with equal authority. The number of teaching elders in each meeting of Presbytery, Synod, or General Assembly is equalized by the number of ruling elders, so that clericalism does not take hold.
Ruling elders and teaching elders have "parity of ministry." It can be said that a presbytery meeting, after being constituted with prayer, becomes a "corporate bishop" to make administrative decisions for a regional set of churches.
Teaching elders hold membership in their respective presbyteries. Ruling elders hold membership in their local church, where they are called to serve by a Nominating Committee, and elected, ordained, and installed by the local congregation.
As to American history ...
James Madison, a major shaper of the United States Constitution, studied under the Rev. John Witherspoon, President of Princeton University (the College of New Jersey). Rev. Witherspoon was not only the only clergyperson who signed the Declaration of Independence; he was instrumental in organizing the first General Assembly of the Presbyterian Church in Philadelphia in 1789. That is how many of the governing principles from Presbyterian polity made their way into the United States Constitution, e.g., representative government, a system of checks and balances, and decision by majority vote with the rights of the minority vote protected.
The Presbyterian Church (U.S.A.) is in full support of the Constitutional principle of the separation of church and state. This priciple is intended to protect the church from the state (and the establishment of any one faith tradition). It is not designed to protect the state from the church's preaching! The founding fathers did not want an established church, such as England had. This idea of the church as a voluntary association, maintained by those who value it (and not the state, through taxes, as it is in England and Europe), has given rise to the preference of Americans to work in thousands of voluntary associations throughout the country, as well as through churches. It has been said by Dr. Ron Stone that church and state must always be kept separate, because faith and politics never can be.