Before I begin this, I'd like to say that it's pretty fucking difficult to summarize Nietzsche's philosophy, and this is not some amazing attempt. For example, I left out entirely Kant's epistemological pessimism, and Schopenhauer's reaction to it, which was a huge influence on Nietzsche. It is important to understand that much of philosophy is a reaction to earlier philosophers, e.g., much of Plato's work is a reaction to the Sophists, and much of Kierkegaard's work is a reaction to Hegel. If you're really curious about this stuff, I'd highly suggest reading Kant's Critique of Pure Reason, and Schopenhauer's On the Will in Nature. Those two works will give you a much better understanding of Nietzsche and his works. As well, I have added some suggested readings at the end of this post, for anyone interested in furthering their understanding of Nietzsche. This post barely scratches the surface of what is an amazing philosopher's work.
First and foremost, Nietzsche is an anti-realist when it comes to morality. This means that Nietzsche denies there is an objective set of moral values. For example, you may think it's immoral to steal a loaf of bread to feed your family, and I may think it is immoral not to. However, the concepts of right and wrong, or good and evil, are nothing more than human created illusions that we attempt to live by, and these concepts do not exist in this world independent of humanity. There are many branches of ethics (e.g., a utilitarian believes in doing things that benefit the most amount of people, while a virtue ethicist would believe that you must live by your virtues in all situations), and Nietzsche would say that they are arguing something that does not exist. What you say is right cannot be proven right, nor can I prove that I am right in my beliefs. Nietzsche’s goal is to free humankind from the false notion that morality is good for them.
Second, it’s important to understand that Nietzsche believes humans are no better than animals. As well, Nietzsche does not believe in the concept of free will. His argument against free will is a bit difficult to explain, but I will do my best. He argues that a being with free will would have to be the cause of himself, or self-caused (causa sui – A is a cause of A), and since we are not self-caused then we do not have free will. If we do not have free will, then we cannot be responsible for our own actions. In fact, Nietzsche argues that we are like animals, going on instinct, but we’ve been given this thing called reason which is not as strong as our instincts. Like Freud’s concept of the human as a battlefield between the id, ego, and superego, Nietzsche believes we have an internal battle between instincts and reason. The will of a human is actually nothing more than the type of person that s/he is, which is based on that individual’s instincts. So, if I am an angry type of person, and I kill someone, then I really cannot be held responsible for that killing, because it was bound to happen due to the type of person that I am. Of course, that doesn’t mean that Nietzsche is saying they should not be punished.
Alright, now on to the most well-known aspects of Nietzsche’s philosophy: his influence on existentialism, as well as his critiques of religion. For Nietzsche, the greatest problem for man is how we justify our lives, and make them meaningful and valuable. He believes that the justification of life through morals and values leads to nihilism, and that is what he sees happening around him. Therefore, Nietzsche believes that we need to justify our lives not through morals, or God, but rather through the highest form of earthly man, Superman. Basically, the role of the Superman is to pursue that beyond the morals, and to suppress the instinct side of ourselves that we discussed earlier. The Superman is the goal of humanity. Rather than setting our goal on things that are not real, i.e., God and morals, it makes more sense to set our goal on creating a Superman of ourselves. Nietzsche believes that Platonism and Christianity distract us from our pursuit of being the Superman, because they rely on reason and faith. Both reason and faith share the feature of having a moral conception of the world. This is seen in Western Civilization as follows: For humanity to enjoy security, they must project their desires onto the world as morals or values. This creates a false world beside this world, and the false world is invented by lies. The false world, morals, values, God, and absolute truth are all words for the same mistake. It’s not just that God does not exist, but that God is a lie, and the word God and morals are two things that mean the same thing. The entire concept has been created by humanity and has become a distraction. The death of God signifies the end of this distraction, whereby mankind can now pursue earthly duties to the real world. It is the false world that creates nihilism, and the death of the lies that lifts us from this state. The Superman lives beyond the concepts of good and evil. The Superman pursues self-mastery, and is able to return to life and rejoin nature (Nietzsche is not painting the Noble Savage portrait of Rousseau).
Nietzsche did not believe that all beings are equal. He believed in an order of rank. The idea of equality was something of the past, something advocated by God, and with God dead we can now see that it too is a lie. Nietzsche believed that there were higher men, and lower men, just as in nature there are dominant members of the pack, and submissive members of the pack (remember, Nietzsche sees no difference between humans and animals). Nietzsche believed that the idea of equality was created by lower men, to bring the higher men down to their level. The lower man makes up most of society, and is the typical man whose only goal is to suppress his instincts. However, the higher man not only possesses great power, creative power, and strength, but also he is able to keep all of these things in control. The higher man has a strong will to power, and can basically suppress instincts. For Nietzsche, the only value of a human is what qualities that human acquires, and the value of humanity exists solely in creating Supermen. With lies such as God gone, the only objective now is to become a Superman.
I Am Not a Man, I Am Dynamite! Friedrich Nietzsche and the Anarchist Tradition - This one I just recommend because it's a fun and interesting read, but it's not necessarily going to give a lot of insight into Nietzsche's philosophy.