I don’t have an exact definition, but my understanding is that it provides vehicles for hire with a driver. Typically the rental is by the hour or the day, rather than by the mile as with a taxicab, and typically they don’t solicit business on the street the way cabs generally do. For example, if you wanted a stretch limousine for prom night or to impress an important business visitor, you’d contact a livery service. Many of them also offer more mundane vehicles for more everyday needs, which I thought might have helped if you hadn’t resoved your dilemma.
Also, consider contacting Rentawreck (rentawreck.ca is the website). They rent older cars and are generally known for having more liberal policies on things like driver age, as well as lower prices.