Taxi or Taxicab "is a type of vehicle for hire with a driver, used by a single passenger or small group of passengers often for a non-shared ride." (see Link)
The word taxicab refers back to the Harry Nathaniel Allen's New York Taxicab Co. and was the short for "taximeter cabriolet". Taxis are specifically licensed to operate within a particular jurisdictions such as the city or a state and operate either via a call or a hail.
Normally the taxis operate on regular production vehicles though there have been several companies that have made cars dedicated to the taxi service. The world famous LTI that produces London's famous black taxicabs and the now defunct Checker that was producing taxis used in the United States up to the 1980s.