It is basically a philosophical movement that tries to address philosophical questions from the viewpoint of individual human existence, focused on the needs, problems and issues of the individual person. Typical themes include alienation, the question of religion and belief in God, anxiety, inauthentic vs authentic living, guilt, subjectivity, action, ethics and morality, and other themes. John MacQuarrie (a very good philosophical theologian) wrote a book about it, and there are other good introductions.
“Nietzsche moved from existentialism to nihilism and went insane.”
Nietzsche, strictly speaking, was not a nihilist. It is more accurate to say he thought there were currents of nihilism in European culture and he wanted Europeans to face up to these currents by renewing European culture with great art, ideas, and visions. I think his view is close to the Greek philosophical tradition in many ways, particularly that of the Pre-Socratics and Aristotle.