Trotskyism is the form of Communism that is based on the ideas of Leon Trotsky. Trotsky described himself as an "orthodox Marxist". This is a different way of seeing the ideas of Karl Marx than the way other communists like Mao Zedong and Josif Stalin saw them. The biggest difference is in Trotsky's idea that there needs to be an international "permanent revolution". A permanent revolution is the idea that proletarian revolution needed to spread in countries worldwide, even where capitalism was not as advanced as only the proletarian revolution could carry out the tasks of the unfinished bourgeois revolutions in these countries. This is different to the Stalinist idea of trying to preserve a single nation's revolution from within. Trotsky also supported a less authoritarian government and more democracy
The largest Trotskyist organizations today are the Reunified Fourth International, International Socialist Tendency, the International Communist League, and the Committee for a Workers' International. Fun fact: Leon Trotsky had an affair with the Mexican artist Frida Kahlo!