What type of character often has a personality that is the opposite of that of the story's main character?

This usually refers to the antagonist , this is the character(s) (or situation) that represents the opposition against which the protagonist must contend. In other words, the antagonist is an obstacle that the protagonist must overcome.

Foil character

Explanation:

Commonly used to highlight and contrast the features of the protagonist, a foil character presents traits that are opposite to the ones of the protagonist, creating relevant parallelism between the two. A foil character can also be an antagonist, but that is not always the case.  

An example of a foil character would be: From the Harry Potter series, Draco Malfoy, whose traits vastly differs from Harry Potter's.

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