Can you summarize the Lydia-Wickham episode in Pride and Prejudice?

Without proper parental supervision, Lydia grows into a flirtatious and egotistical young woman. She is, in today's language, "boy crazy" and indiscriminate in her affections. Lacking social sophistication, she marks herself as an easy target, especially to the military men who are stationed near her home.

When Lydia goes to Brighton to be near the regiment, she is even less able to regulate her impulses than when she is among her sisters. It is not surprising that she falls prey to Wickham's devices, seemingly eloping but not marrying until Darcy intervenes.

This episode occurs just as Elizabeth and Darcy are seemingly reconciling. Elizabeth is appreciating Darcy's superiority in taste and status, as she failed to do at Netherfield. Austen seems to include this difficult interlude not only to extend the drama of the novel (or to show how beneficent Darcy is) but also to offer a greater appreciation of the marriage Elizabeth and Darcy achieve. Lydia is one example of the dangers that await women without wealth who fail to act with caution. Elizabeth's appreciation of Darcy is not based on the sentiment that drives many movie versions of the novel but on a deeply felt respect for what it means for him to associate himself with her family.

Assuming you are talking about the elopement of Wickham and Lydia, this is what happened. Lydia is basically a very thoughtless creature who fantasizes about being married and wants to marry an officer in uniform. Wickham is an immoral, irresponsible, and equally worthless individual who takes advantage of her looseness and elopes with her as part of his being an adventurer. In the end, Mr. Gardiner, the person in charge of Lydia at the time she was staying there, finally finds them, and they are married off thanks to money that was actually given by Darcy to secure a dowry and try and save her reputation.

What is the significance of the Lydia-Wickham episode in Pride and Prejudice?

In this novel, Jane Austen explores the theme of how both pride and prejudice can blind people and lead them to draw the wrong conclusions. While we typically see Darcy as "pride" and Elizabeth as "prejudice," both characters display both traits and both make mistakes. If Elizabeth prejudges Darcy based on surface appearances, including his rude comment about her not being worth dancing with, Darcy prejudges Jane on surface appearances, believing because of her outward coolness that she really doesn't love Bingham.

Elizabeth's great initial blunder is to believe the charming Wickham when he tells her stories of being cheated and wronged by the proud Mr. Darcy. However, when Mr. Darcy tells his side of the story in a letter, Elizabeth, as she expresses to Jane, is then inclined to believe Mr. Darcy. At this point we don't know whose story is true: it's a classic situation of one person's word against another.

The Lydia-Wickham situation thus becomes very important because it demonstrates who is the truth-teller and who is the liar. Wickham shows through his actions—running off with and "ruining" a young girl with no intention of marrying her—what a wicked person he is. We now have every reason to believe Darcy's version of events. Further, in going to such efforts and expense to arrange the marriage of Wickham and Lydia, Darcy shows he is a person of integrity and that he is genuinely in love with Elizabeth. Wickham exposes himself as a self-indulgent, selfish loser and Darcy reveals his depth of character.

Modern audiences "know" how the story ends, but if we project back into the minds of early readers, one can imagine that doubts would linger about who was telling the truth: Darcy or Wickham. The dramatic action of Wickham seducing Lydia puts all that to rest.

Of course, this episode also drives the plot forward. It gives Darcy the chance to demonstrate the sincerity of his love for Elizabeth and to show that he has grown beyond judging her on the basis of her family's imperfections. It also kills forever any lingering feelings Elizabeth might have had for Wickham. 

It leads as well to Mr. Bennet's reformation. No longer will he be so careless and uninvolved in his daughter's lives. He realizes and regrets that he has let his younger daughters rove and flirt too freely. He knows what an extremely fortunate escape from disgrace his family has had in the marriage of Lydia and Wickham.