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––you mom could compel you to finish your homework by threatening not to feed you until it's done. A compelling argument compels you to agree with its logic––it's irresistible. That cute skirt you’ve been eyeing for months is now 75% off? The price is a compelling reason to buy the skirt.