I guess, "Vertretungsprofessor" is a rather European or even German concept. When a professor cannot teach, someone else with equivalent qualifications is substituting for her. This can be rather planned for professors on sabbatical or for the period after a professor retires and before a new one has been chosen. Sometimes you can even apply to substitute, because there will be an official advertisement. Most of the time, the person substituting for a professor will be contacted directly. So one should have a good network, have an uptodate website, and be prepared to teach something new on short notice. There are also rather unplanned occasions, when a professor get's seriously ill or even dies, when a professor applies for some kind of sabbatical that might or might not be accepted, or when a professor accepts a new position at another university on short notice.
Several concepts exists how substituting works: Other faculty members or staff members teach single courses as additional teaching load to their regular classes. Sometimes a course is taught by an external or internal lecturer who is paid for this course as an adjunct lecturer. So the teaching load of the professor is split up between several lecturers. It's not uncommon in Germany that adjunct lecturers get paid rather symbolically, the paiment is a few dozen Euros per hour taught, i.e., excluding time needed for preparation, grading assignments, or answering students' questions.
The fact that another researcher -- who in principle would be eligible for professorship -- acts for the professor on sabbatical, is rather special. The substitute does not only do the teaching, but also overtakes all duties and responsibilities like supervising masters' theses, grading final exams, and attending meetings. However, most of the time the substitute gets paid according to the PostDoc scale -- i.e., according to the current status of the substituting person. The University of Konstanz pays acting professors according to the salary scale of the professor who is substituted for. This is rather unusual, I guess.
However, I'm still a PostDoc, I'm not a regular Professor and I'm not allowed to use this title. In some occasions a researcher substituting for a professor might even be allowed to use the title during the time they substitute.