Trotskyists regard themselves as working in opposition to both capitalism and Stalinism as embodied by the leadership of the Soviet Union after Vladimir Lenin's death. Trotsky advocated proletarian revolution as set out in his theory of "permanent revolution", and believed that a workers' state would not be able to hold out against the pressures of a hostile capitalist world unless socialist revolutions quickly took hold in other countries as well. This theory was advanced in opposition to the view held by the Stalinists that "socialism in one country" could be built in the Soviet Union alone.
1) My quote is from wikipedia.
2) Stalin was a socialist who wanted socialism in one country. Which is to say, National Socialism. He had an alliance with Adolf Hitler (another National Socialist).
3) Trotsky thought Stalin didn't go far enough in his communist beliefs. The folks who think Hitler was "right-wing" are folks who were further Left than Stalin. That's Trotsky.
4) The Fourth International carries the banner of Communism today and they are far more active and influential than people think. They promote Trotsky's belief's.
People shouldn't pay attention to Marx. Marxism is truly dead. But Trotsky, Gramsci, and the Frankfurt School are (sadly) alive and kicking.
Socialism shows its effects in the USA, because there is a demand for labor that is not filled it own natives.