What is Ecclesiastical Jurisdiction?

Legal Definition
Ecclesiastical jurisdiction in its primary sense does not signify jurisdiction over ecclesiastics ("church leadership"), but jurisdiction exercised by church leaders over other leaders and over the laity.

Jurisdiction is a word borrowed from the legal system which has acquired a wide extension in theology, wherein, for example, it is frequently used in contradistinction to order, to express the right to administer sacraments as something added onto the power to celebrate them. So it is used to express the territorial or other limits of ecclesiastical, executive or legislative authority. Here it is used as the authority by which judicial officers investigate and decide cases under Canon law.

Such authority in the minds of lay Roman lawyers who first used this word jurisdiction was essentially temporal in its origin and in its sphere. The Christian Church transferred the notion to the spiritual domain as part of the general idea of a Kingdom of God focusing on the spiritual side of man upon earth.

It was viewed as also ordained of God, who had dominion over his temporal estate. As the Church in the earliest ages had executive and legislative power in its own spiritual sphere, so also it had judicial officers, investigating and deciding cases. Before its union with the State, its power in this direction, as in others, was merely over the spirits of men. Coercive temporal authority over their bodies or estates could only be given by concession from the temporal ruler. Moreover, even spiritual authority over members of the Church, i.e. baptized persons, could not be exclusively claimed as a right by the Church tribunals, if the subject matter of the cause were purely temporal. On the other hand, it is clear that all the faithful were subject to these courts (when acting within their own sphere), and that, in the earliest times, no distinction was made in this respect between clergy and laity.
-- Wikipedia
Legal Definition
Jurisdiction over ecclesiastical cases and controversies; such as appertains to the ecclesiastical courts. Short v. Stotts, 58 Ind. 35.
-- Black's Law Dictionary