In "Pride and Prejudice" Darcy refuses to dance with Elizabeth. At the next ball it is the reverse. What is the significance of this irony?

The irony of Elizabeth's actions hits Darcy over the head.  He is not used to women refusing him, he is accustomed to women falling all over him because of his social position and financial status.  Elizabeth, a gentleman's daughter, but a pauper compared to Darcy, has the nerve to reject his attention toward her.  This is unthinkable for the prideful Darcy at this point in the story.

But what is particularly interesting is the way that Austen writes these characters, giving Elizabeth Bennett a personality that allows her to reject a rich, rich man even though her family is desperate for a good solid marriage from her or Jane.  Elizabeth 's integrity, honesty and devotion to the truth are the main reasons that Darcy falls head over heels in love with her.  She does not grovel at his feet, even though, she, more than Caroline Binghley, needs Darcy's money and position. 

It is very ironic for Darcy, then, that Elizabeth sticks to her principles, rejecting him for his arrogance,  even though Darcy is very rich and could be a savior to her family's situation.