To charm means to cause joy or to delight. A person is called charming if they are likeable and friendly or easy to talk to.
Ex: She charmed everyone she talked to at the party with her bubbly personality and easy conversation.
Ex: He had a hard time growing up always being compared to his charming and athletic older brother. He tended to be more argumentative, especially about politics.
Charm refers to how a person affects those around them, or more precisely, how they delight them. However, a charm (especially a lucky charm or magic charm) can also refer to some small object used to keep away bad luck or misfortune. While they are generally agreed to be nothing more than superstition, some people find comfort in a charm.
Ex: The revealing newspaper article is just the result of another employee falling for the reporter's charms and saying more than he should have done about the company's inner workings.
Appeal means the power or ability to attract interest or admiration, as in a person or a thing that has appeal. It can be used both as a noun and as a verb. Appealing would be the adjective.
Ex: Although Frank does not usually eat pork, he mentioned that the way you present food makes it so appealing that he wants to try it.
Ex: Does going to an amusement park on a holiday appeal to you, or do you find that idea repulsive?
If you appeal a decision, you are asking a person of higher authority to review your case or request to overturn the decision that was made earlier.
Ex: Sally planned to appeal the dean's decision to have her withdrawn from classes due to her illness before the semester's end.
Ex: The state supreme court has agreed to hear the accused robber's case on appeal, as long as his lawyers file the appropriate documents in time.