The "overman"
The overman (superman, or Übermensch in German) is a problem for the many non-overmen that exist in the world. Nietzsche dreamt up this idea (and he did dream it up; it was not the product of some kind of publically-disclosed logical process), and it has, in our creative classes, only served to create a contest as to who can commit spiritual or actual suicide the fastest, or at the very least, die in the most heroic way possible.
The overman is Nietzsche's solution to the "death of
God" (as if
God would simply die if we forgot him, but I digress). He posits a future where belief in
God seems ridiculous, priests are mendacious, and the way we think has simply changed to the extent that there is no place for
God in it. If you want the truth, philosophers have simply through their influence made a mockery of truth, creating one philosophy that doesn't hold up to careful examination after another, filling peoples' heads with incorrect notions, and Nietzsche sees this process as somehow having killed
God.
How the overman fits into this is that Nietzsche saw the overman as someone who overcomes himself and lives by the Will to Power, which Nietzsche may or may not have seen as the fundamental principle of nature when said principle was detached from human pretense. The superman, most importantly, was "beyond
good and
evil", meaning that he had dispensed with things that hold societies together (like morality) and lived as his desire carried him.
Unfortunately for the overman, one can not overcome oneself without
God, and so the overman is caught in quicksand trying to destroy who he
really is to become what he
wants to be. In this sense, the doctrine of the overman is fundamentally suicidal and self-hating, which are accusations Nietzsche levelled at Christianity itself.