Lydia is a disgrace in two regards. The first is because of her continual flirtatious and indecorous manner. She is always flirting with the military men stationed in Meryton and prompting her sister, Kitty, to follow suit. This is a disgrace because it was not socially correct for young women to be so forward in their behavior toward men. This is one of the things that Darcy thought so bad that he advised Bingley against pursuing his love for Jane. Darcy advised that a family with a daughter so uncontrolled could only involve Bingley in social scandal of one sort or another, as would the whole family.
"If you were aware," said Elizabeth, "of the very great disadvantage to us all, which must arise from the public notice of Lydia's unguarded and imprudent manner; nay, which has already arisen from it, I am sure you would judge differently in the affair." (Chapter XVIII of Volume II / Chapter 41)
The second is because, after forcing her father to allow her to go to Brighton with the Forsters, she eloped under cover of darkness with Wickham. It turns out that Wickham left in secrecy at night to escape debts he had amassed. Lydia begged him to let her go along. Notwithstanding, Lydia vanished in the night with a man she was not married to and who was neither in her family nor a family friend who might be called her protector. The assumption was that they had eloped together with designs in marrying in Gretna Green in Scotland where marriage laws were more lenient than in England. This brought almost irrevocable social disgrace upon the family that would end in their rejection from society and in the dissolution of the other girl's chances to marry anyone in the upper class or anyone with any wealth or connections of any kind, except Darcy saved them from this fate by arranging a marriage.
Lydia ran off with Wickham. At that time period it was not acceptable for young ladies to be alone with men like that. When they were found livining together alone it disgraced her and as a consequence her whole family. They were quickly married to try and cover up the indescrtion but the damage was already done.