I'm glad you ask.
His genealogical approach to ideas has been very influential on philosophers like Foucault.
He also marks a return to a more artistic and poetic style of philosophy (obviously influenced by his deep relationship with ancient greek philosophy) and breathing some life into what can be a very dusty discipline. I think this helped open the door to many post-structuralist philosophers like Derrida, Baudrillard, and Deleuze and Guattari.
I like his thought experiment of eternal recurrence. Live your life like it will be repeated over and over eternally.
His aphorisms are SO quotable and fun to unpack. "Be careful fighting monsters, lest you become one." “The criminal is quite frequently not equal to his deed."
A great deal of his project is to reject self-effacing/self-denouncing ideology (what Zizek might call the Big Other) and to embrace a self-affirming life. Become who you are (rather than reject who you are). His ethics are similar too (and I would argue draw heavily from) Aristotelean nobility ethics. Be good, not for the sake of others, but for the sake of your own greatness (especially greatness of spirit which manifests through things like generosity, liberality, etc.) and is the opposite of mean spirited. Embrace the "will to power," which is at the heart of all human action anyway, rather than be hypocritical about it.
His idea of self-becoming is also quite interesting. He suggests a certain level of determinism--like you can't change who you are or even change your behavior very much. But what you can do is embrace who you are (including what some might consider dark or base or animalistic) and tell a different story about your past actions. His thinking is deeply humanist. I also find his ideas to be quite liberating.
His philosophy is sometimes described as "nihilistic," but this is not accurate. It's not that he thinks there is no meaning or truth. It's that he believes there are a multitude of meanings and truths. His works invite you to cast off tradition and become something new.
Nietzsche also has a cool twist on Christ's admonition to "love your enemies."
He says "love your enemies" but to love them "as enemies."
Love your enemies because they bring out the best in you. Nietszche taught that you should judge a person not by her friends, but by her enemies. The stronger your enemies, the stronger you are.“[Y]ou must be proud of your enemy… For the worthier the enemy, O my friends, you shall save yourselves…”
“And when I want to mount my wildest horse, it is always my spear that helps me up best, as the ever-ready servant of my foot: the spear that I hurl against my enemies. How grateful I am to my enemies that I may finally hurl it!”
“Learning from one’s enemies is the best way toward loving them; for it makes us grateful to them.”
You should love your enemy for providing adversity and challenge in your life and giving you an opportunity to exhibit your greatness.
The flip side of this is that you should learn to hate your friends for the very things that make them friends. Just by being kind, friends may coddle your ego or indulge your weaknesses. You should recognize this and, at least to a degree, hate them for their kindness and encourage your friends to be hard.
“But if you have a suffering friend, be a resting place for his suffering, but a hard bed as it were, a field cot: thus will you profit him best.”
This concept of loving-ones-enemies-as-enemies ties in to his concept of the will to power and the central place of conflict in human nature. This resonates with me as an attorney. My job is fighting one conflict after another.
This also ties in his elevation of beauty and individualistic artistic expression over philosophy and conventional ethics. When he writes about overcoming good and evil, I believe this is what he is point too. Live your life beautifully and artistically rather than merely according to what is "good." He also encourages each person to write their own moral code--because that is more interesting than following someone else's.
His concepts of Dionysian and Apollonian are also really useful. Both are important; however, the Dionysian is often demeaned. He elevates the Dionysian in a way that is not often seen in philosophy (which tends to be very stodgy and Apollonian).
I could go on.
