Webster 1913, under "Corporation," tells us that:
A corporation sole consists of a single person, who is made a body corporate and politic, in order to give him some legal capacities, and especially that of succession, which as a natural person he can not have. Kings, bishops, deans, parsons, and vicars, are in England sole corporations. A fee will not pass to a corporation sole without the word "successors" in the grant. There are instances in the United States of a minister of a parish seized of parsonage lands in the right of his parish, being a corporation sole, as in Massachusetts.
It is the traditional opposite of a corporation aggregate, which has more than one owner. In the olde English days corporations had to be named after their membership—a rule still reflected today in names like "The President and Fellows of Harvard College"—and a corporation sole was named after a single person, like "The Archbishop of Boston." Business corporations with only one shareholder are not considered to be corporations sole, as the corporation is the business itself and not the person owning the business.
Religious organizations account for most of the corporations sole that exist today. The Roman Catholic Church, for instance, sets up a corporation sole for each of their bishops. This allows each bishop to "own" the church property within their diocese, without giving the bishop personal ownership of that property. This also allows the property to pass from bishop to bishop relatively easily: when one bishop dies or resigns, the Church can transfer the corporation sole to another bishop.
Some monarchs are also legally established as corporations sole (according to Wikipedia, the Queen is a corporation sole in New Zealand).
In the United States, corporations sole are only usable by religious entities, and they get special exemptions from income tax liability under Section 501(c)(3). Occasionally, smartasses try to dodge the IRS by setting up a nominally-religious corporation sole and running their affairs through it; the IRS doesn't like this and usually audits the living daylights out of anyone who gets caught doing it.