There are many takes and definitions of existentialism but in it's essence is that life has meaning. People find this meaning in life by living therefore there are no gods or other people that make your density. Humans who believe in existentialism think that this meaning separates humans from other animals on Earth. Gregor when he is human never really finds this meaning to his life, well the true meaning anyway. All of Gregor's adult human life was dedicated to help his good-for-nothing family pay off their debt. Once that debt is payed off the text never says what Gregor has planned for the future except that he is going to quit his hellish job. It also doesn't help that he has no friends to speak of and his family is a bunch of …. jerks who never really cared about him. Since he has almost no meaning in his human life it is not surprising that he dies quite rapidly once his one purpose, keeping his family in the lap of luxury, is taken away from him. In existentialism it is also believed that if you commit suicide you have chosen to have no meaning. Gregor at the end of the story does commit a form a suicide by starving himself to death. Which by existentialist terms means that Gregor has decided to have no meaning and therefore no life.
Nihilism is where existence is pointless. Just when you thought it could get any more depressing, it does! It's quite hard to explain nihilism because its believers believe in nothing which apparently includes not believing in writing down exactly what they don't believe in. The only way that it relates to existentialism is that they both don't believe that higher power is controlling them. Yet again nihilists believe in nothing so does that mean that as soon as someone doesn't believe in something then they have something in common with a nihilist? Gregor isn't really a nihilist until the very end of the story where he commits suicide because if he believed that he had a purpose to live for then he wouldn't have committed suicide. Gregor's purposes in life turn more and more pitiful as his life as a bug crawls on and soon his purpose in life is just to live and stare out a window. But then again it quite easy to see why Gregor ended up in this sad state, I mean, his family left him for dead, once Gregor had in fact passed on he was thrown out with the garbage and his family caught a trolly into town to celebrate. Even if a person had a very loving, accepting family being a bug is still a pitiful existence for a human because we figure that we are above all other life forms.