Ok, well, you chose one of the most difficult philsophers to ask for a simplification (other than possibly Heidegger). Here goes:
Art: Apollo/Dionysus represent opposing sides of Art and Tragedy. Nietzsche liked Dyonysian art and believed that the current German art scene most represented the classical Dionysian ideal first started by the Greeks.
Morality/ Death of God: When Nietzsche says that God is dead, what he means is that our entire concept of “God” is dead. The Christian concept of God as an all-powerful, all-knowing, morally good, caring, Supreme Being is dead, according to Nietzsche. But God no longer listens to us, nor knows how to respond. This lack of communication between us and God is the cause of the decline of theism (Beyond Good and Evil, Sec. 53). New forces have taken the place of God in our lives, forces such as money and science have provided a new faith for us in an increasingly materialistic society. We have become religious atheists, we no longer have God, but we practice religion. But religion involves sacrifice, and that is why God is dead.
Nietzsche was concerned that the death of God has lead to a pervasive state nihilism. By sacrificing God in the name of asceticism, asceticism has lost its purpose and meaning to life. Nietzsche argued that one way in which we attempted to fill the space left by God’s death is through secular science. Since we have traded God for science, we now worship science as our new God. Science has occupied the throne on which God used to sit. Science has become the new ascetic faith, populated by theoretical entities and forces. But unlike theism, science ascribes no meaning to life’s happening. It only provides the means by which life happens; it provides the how, not the why.
For Nietzsche, the alternative to nihilism and asceticism lies in the positive affirmation of life.
The Eternal Return: This is Nietzsche's greatest contribution to Philosophy and I fear I do not do it justice by simplifying it but heregoes: It means that everything happens over and over into infinity. It is also the one aspect of Nietzsche’s philosophy that is the most elusive and often misinterpreted. Perhaps most misunderstandings arise from interpreting the eternal return as a physical theory of recurrence. But I believe Nietzsche intended the eternal return as a philosophical theory that transcended mere physical theories. A physical theory describes how one process works in the universe, but the eternal return describes how all processes and events work in the universe. It gives order to the universe.
Nietzsche’s argument for the eternal return is based on logic, not any particular physical facts. It does not presuppose anything physical, but only that there are a finite number of combinations of physical things in the universe and there is an infinite amount of time. Using this formula, Nietzsche argues that all possible combinations of events will occur and all possible histories of the universe will occur an infinite number of times. This means that every person’s possible history will occur infinitely often. Every possible biography of your life will occur. Nietzsche argues for destiny or fate. You will live every possibility, but not necessarily in this specific life. But through the eternal return all possibilities will be realized. By accepting this, one realizes that free will is an illusion.
The overman/superman: The overman is the one who finds the truth in the eternal return and finds comfort in it. Since the overman is the one who is free from the restraints of moralistic oppression and finds value in life and its happening, there is no greater joy than being able to relive that joy for eternity. And the overman is able to live without regret and disappointment because he realizes that he is living out every possibility for eternity.
There is much, much more and this is only the tip of the iceberg, but it should be enough to give you an introduction.
I have a degree in Philosophy