Why was Elizabeth's visit to Derbyshire important?
Elizabeth Bennet visits the county of Derbyshire with her aunt and uncle Gardiner. While there, her aunt tells her that the county is home to Pemberley, Mr. Darcy's grand estate. Because they learn that the family is away from home, the trio decide to go and tour the home (a relatively common practice with large estates like this one). Here, Elizabeth hears from Mrs. Reynolds, a servant who has known Mr. Darcy since he was a child, that she's "never had a cross word from him in [her] life" and that she doesn't view him as proud at all. This clashes very strongly with Elizabeth's earlier impressions of the man, and begins to confirm the goodness she glimpsed in his post-proposal letter to her from when she was in Hunsford.
Further, when Darcy returns early and bumps into Elizabeth and her family, he addresses her with such "civility"; she'd never seen "his manners so little dignified, never had he spoken with such gentleness as on this unexpected meeting." He is kind and cordial to her aunt and uncle, and he expresses a wish of introducing Elizabeth to his sister. It becomes obvious fairly quickly that he still loves Elizabeth, and she later says that it was around this time that she began to love him.
Finally, the fact that Elizabeth is staying at Lambton and that Darcy can call so easily means that he can be the first person she sees after news of Lydia's elopement reaches her. Thus, he is able to step in and resolve the situation before it becomes a really big scandal for the Bennet family. It is, later, Elizabeth's wish to thank Darcy for his involvement that once again prompts him to share his unaltered feelings for her. And, by this point, she returns them.