the Master is a Time-Lord so, like the Doctor, he is able to regenerate when he is close to death.
the Master was already on his thirteenth incarnation, and dying when he used the Trakenite power source to take over a Trakenite's body. He later used another method to take over the body of a human. The Time Lords later granted him thirteen more lives. There's a lot more to it than that, but going into specifics would start an argument.