Your question is a little bit vague because Lydia never makes any direct demands on Darcy. However, you could be referring Lydia's behavior. Lydia's behavior in running off with Wickham makes Darcy commit to paying Wickham's debts, paying for Wickham's commission into the army, and even making a thousand pound settlement on Lydia, all so that Wickham will marry Lydia. Darcy felt obliged to do this for two reasons: 1) He wanted to rescue Elizabeth's honor, which would have been ruined by Lydia's behavior; 2) He was too reserved to make Wickham's poor character widely known, and felt responsible for Wickham's treatment of Lydia (Ch. 10, Vol. 3).
You may also be referring to what Mr. Darcy requested of Lydia, or asked her to commit to. Darcy requested that Lydia keep his involvement in their marriage a secret. Out of humbleness, he wanted the family to believe Lydia's Uncle Gardner was monetarily responsible for blackmailing Wickham into marrying Lydia (Ch. 9, Vol. 3).
Mr. Darcy finds out where Lydia and Wickham are living, pays off Wickham's debts, pays for the marriage, and gives Wickham his commission but he does this out of love for Elizabeth even though she rejected his proposal.