In the early nineteenth century, fancy people living in London during the social season could go to weekly balls held at a private club called Almack's. Velvet ropes had nothing on this place—it was so exclusive and so strict that they would allegedly turn away dukes for not being dressed right. But the stakes were high: it was one of the only places for young men and women to find eligible marriage partners. In fact, it was so notoriously designed just for that reason that it was called the Marriage Mart (source).
The people living in and around the small country town of Meryton aren't spending much time at fashionable balls, but they are interested in getting married—and getting the news, and playing cards, and dancing, and generally doing other things than sitting around in a drawing-room after dinner with their families. Their balls might not be as exclusive as Almack's, but they were just as important. Look at how Austen describes the first ball at the Lucases':
The evening altogether passed off pleasantly to the whole family. Mrs. Bennet had seen her eldest daughter much admired by the Netherfield party. Mr. Bingley had danced with her twice, and she had been distinguished by his sisters. Jane was as much gratified by this as her mother could be, though in a quieter way. Elizabeth felt Jane's pleasure. Mary had heard herself mentioned to Miss Bingley as the most accomplished girl in the neighbourhood; and Catherine and Lydia had been fortunate enough never to be without partners, which was all that they had yet learnt to care for at a ball. They returned, therefore, in good spirits to Longbourn, the village where they lived, and of which they were the principal inhabitants. (3.6)
What's going on here is a lot more than dancing with boys, although that's definitely important. The balls are a way to form, strengthen, and test community bonds. And if there's one thing we want you to know about Austen, it's that she's all about the community. What makes a good community? How does news travel? How do people act in communities vs. how they act in private?
You could even say that the Lucases' ball and the Netherfield ball bookend the novel's first act: the Lucases' ball introduces Darcy and Bingley, setting the plot in motion; and the Netherfield ball convinces Darcy to book it out of town, setting up all the complications of the second act. Yep. Pretty important.