The Frankfurt School was a group created at the Institute for Social Research at Goethe University Frankfurt in 1929. The Frankfurt School was opposed to capitalism but didn’t support the form of Communism that Vladimir Lenin created in the Soviet Union.
The Frankfurt School thought that Karl Marx got a lot right but also some things wrong so they decided to fix classical Marxism by mixing Marx’s ideas with the ideas of Sigmund Freud and Georg Wilhelm Friedrich Hegel. The Frankfurt School thought that, instead of using violence, the best way to fight capitalism was to explain how the culture it created made things worse for a lot of people. So Frankfurt School intellectuals like Jürgen Habermas, Theodor Adorno, Max Horkheimer, Herbert Marcuse and Erich Fromm created Critical theory to do that.
The people who started the Frankfurt School ran away from Germany to escape Adolf Hitler after he took over and instead came to Columbia University. The far-right has a lot of conspiracy theories about the Frankfurt School.