a radical is anyone with highly unorthodox political views.
a socialist is one who adheres to socialism.
in some countries like ours, socialism is very uncommon as a political belief, so people who support it are radical. however, in some countries they would not be radical at all. also, in our country, people who adhere to different unusual political tenets (like Nazism) would also be radicals.
socialism is communism. communism is an economic principle based on the principle of sharing--people make what they can, put it in the middle, and take what they need--but many people think its synonymous with totalitarian government. people disagree about the difference between socialism and communism, but you can almost always get away with using them interchangeably.