A compelling situation or argument is one that makes you feel like you really need to do something or believe it's true.
You might hear someone describe a movie as "compelling" because they were completely drawn into the story and characters. You could also say that a news article is compelling if it presents some really strong evidence for its point of view, so much so that you start to think about changing your opinion on the topic. Compelling can even be used to describe a person - maybe someone has such an amazing talent or achievement that they're truly inspiring and motivating.
present participle and gerund of compel
