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

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.

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.