In terms of the history of the world, the most important influence that Nietzsche had on the 20th century was his influence on Adolf Hitler. Hitler was strongly influenced by Nietzsche in his creation of the Nazi ideology.
It must be said that most scholars feel that Hitler’s understanding of Nietzsche’s work was not very good. They feel that Nietzsche’s sister twisted his works into ideas that were congenial to the Nazi ideology. Hitler used, for example, Nietzsche’s idea of the “superman” to justify his own ideas. He saw himself as a superman of sorts who could reject conventional morality because it did not fit in with his desires.
Nietzsche also had much more benign impacts on the 20th century. His rejection of conventional morality and conventional thinking struck a chord with many people. These included artists such as George Bernard Shaw and thinkers such as Freud and Foucault.