Warning: Wikipedia contains spoilers
A teenager, as we would call him now, Mort's personality and temperament make him rather unsuited to the family business. Thus, his father takes him to a local hiring fair, hoping he'll be picked up as an apprentice by someone; not only will this provide a job for his son, but it will also be someone else's problem to knock him into shape. As time ticks by, all of the prospective masters pass him up, but Mort stubbornly holds out to the end, refusing to leave until midnight has passed. Just before the clock-bell rings for the 12th time, a man concealed in a black cloak arrives on a white horse. He says he is looking for a young man to assist him in his work and picks up Mort. The man turns out to be Death, and gives Mort an apprenticeship to his "job" of ushering souls into the next world (though his father thinks he's been apprenticed to an undertaker).
It is believed that Death chose Mort partly because Death himself has the nickname Mort (he is called this by the other members of the 4 horsemen), Mort being the French word for death, and the origin of several English words (e.g. mortal, post mortem).
The pressure of the job forces Mort to make a few mistakes, but like all good heroes, he grows some spine, gains some self control, challenges death to a duel and waltzes away with the girl...
He marries Death's adopted daughter Ysabell and they live relatively happily ever after as the duke and duchess of Sto Helit.
Their storyline (and that of their daughter Susan) is continued in the later Discworld novel Soul Music.