Compelling means attractive, or irresistible, or really, really convincing. You know your argument for backpacking across Europe is compelling when your parents not only let you go but also pay for all your expenses.
To compel is to drive or force into action –– the way the fear of being grounded might compel you to come home before your curfew. A compelling argument compels you to agree with its logic –– it's irresistible. If you decide to run for class president, you'll need to write a compelling speech about why students should vote for you.