The poem begins with the first two lines delineating the theme of the poem, which is basically "Death, you are not the hot shit you think you are." Then John Donne goes on to say that even though Death thinks he is killing people he really isn't, and that he (Donne) himself cannot be killed either.
In lines 5-8 he talks about how restful and pleasureable it is to sleep, which are but "pictures of death," so the real experience of dying must be even more peaceful. The "best men" are removed from life sooner, and their souls are set free. Then Donne continues by telling Death that he really cannot kill anybody anyways, unless helped by disease, war, poison, fate, or "kings, and depserate men." So he says, Death does not have the right to be proud when he can't even choose who he kills. Finally he ends the poem by saying that dying is really just a "short sleep" before the rest of our eternal life (John Donne was a very religious man who firmly believed in Heaven) and so in the end, it is Death, and not us, who dies.