Appeal means "to ask, or address." If you appeal to someone's better nature, you're asking them for mercy. If a shirt doesn't appeal to you, you could also say it doesn't "speak" to you, or more simply, you don't like it.
Appeal can also be used as a noun to refer to a request, as in "his parents ignored his appeal for a later curfew," or to refer to something's attractiveness or desirability, as in "we all agreed on the appeal of a tropical vacation." In judicial contexts, appeal means "to call upon a higher court to review a lower court's decision." If a lawyer appeals a court's conviction of her client, she's asking a higher court to throw the decision out. This word descends from the Latin appellare, "to address, call upon."