The ability to regenerate into a completely different person, with not only a different face but a different personality was first introduced by the show producers when William Hartnell has to leave the show. In a very clever approach, they decided to give The Doctor the power of change into someone else when he is close to death and since then, the transition from one actor to another has become part of the show and totally immersed into the plot. Despite all the different incarnations of the Time Lord, they are all the same man only showing different aspects of a very complex character.
From 1963 to 2016, from William Hartnell to Peter Capaldi, thirteen actors had given life to The Doctor and due time traveling being part of the plot, sometimes different incarnations of the Doctor meet each other. The Last incarnation, the Twelfth, showed up first time in a cameo at “The Time of the Doctor”, when Peter Capaldi took the role after Matt Smith exit.
(*) Image credits to BBC.