Geneva had already voted to live by the gospel and expelled its bishop before John Calvin arrived, which matters, because he never governed the city. He was a salaried preacher, an immigrant without citizenship until near the end of his life, and he was thrown out in 1538 and only invited back in 1541 on his own terms. Those terms were the Ecclesiastical Ordinances, which set out four offices for the church: pastors to preach, doctors to teach, elders to oversee conduct and deacons to care for the poor.
The instrument that made the arrangement famous was the Consistory, a weekly meeting of the pastors with a body of lay elders drawn from the councils. People were summoned before it over absence from sermons, quarrelling with neighbours, dancing, fortune telling, beating a wife, drunkenness, and marital disputes of every kind, and the surviving registers are among the richest sources anywhere for how ordinary sixteenth century people lived. Its usual outcome was admonition and reconciliation. Its ultimate sanction was exclusion from communion, and whether the church or the city council held that power was fought over for more than a decade before the pastors won it in 1555.
The result was not a theocracy in the modern sense, since the civil magistrates kept every coercive power and frequently overruled the ministers. It was something newer: a church with its own disciplinary jurisdiction, exercised by laymen and clergy together, inside a republic that was not itself the church. That model, more than any doctrine, is what the Reformed tradition exported.