The Higher Self is the aspect of a character which "knows better". More specifically, however, the Higher Self is the aspect which rises above whatever is going on in the plot and can see the situation in a way that's removed from emotional or melodramatic entanglement.
Frequently in animation and comedy, this is portrayed as the angel on one shoulder that advises goodness. Drama personifies it in a number of different ways. One common way is to have an older version of the character do voice-over narration commenting on his younger-self's actions. Another way it is personified is to have a character that represents the higher self and another character that is more involved in the passions of the moment. Together, they depict the parts of one whole person. This would be similar to the way The Spock is used in a Power Trio.