Left wing and liberal are the same thing. The terms generally mean being open to using the power of the state to do good things for the population -- like reducing unemployment, regulating food safety, or protecting the rights of unpopular minority groups.
Right wing and conservative are the same thing. Since approximately 1980 those terms have also been synonomous with Republican. They generally opposes using the power of the state to do anything beyond defense and infrastructure. It opposes regulations on anything, all forms of taxation, or any deviation from traditional social norms in the belief that society benefits the most from minimal government.
Fascism is a form of government headed by a strong dictator not bound by law or public opinion which generally attempts to enforce strict order on society (as in the book/movie 1984). It is neither liberal nor conservative.
Liberals often accuse conservatives of being fascists but this is not strictly true. True conservatives oppose a strong central government. The charge is often made because of the emphasis that conservatives place on social conformity and public order.
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