Presbyterianism
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Presbyterianism is a Reformed (Calvinist) Protestant tradition named for its form of church government by representative assemblies of elders. Though there are other Reformed churches that are structurally similar, the word Presbyterian is applied to churches that trace their roots to the Church of Scotland or to English Dissenter groups that formed during the English Civil War.
John Knox
The Ordination of Elders in a Scottish Kirk, by John Henry Lorimer, 1891. National Gallery of Scotland.
Celtic cross draped for Easter at a Presbyterian church
Ecclesiastical polity
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Ecclesiastical polity is the government of a church. There are local (congregational) forms of organization as well as denominational. A church's polity may describe its ministerial offices or an authority structure between churches. Polity relates closely to ecclesiology, the theological study of the church.
An Anglican deacon, bishop and priest. Priests are usually former deacons in episcopal polity.
The ordination of Methodist Bishop Francis Asbury, 1784.
Cathedral churches like St. Andrews were incompatible with the presbyterian polity taught by John Knox. This statue stands in St. Giles, still called a cathedral despite no longer serving as an episcopal seat.