Delphini "Delphi" Diggory is Voldemort's secret daughter (wow, the Potterverse is turning into a soap opera). She wants to resurrect the Dark Lord—not necessarily to bring evil to the world, but just because she wants to meet her dear ol' dad.
It's actually kind of sweet…in a completely evil way.
Voldemort often used Harry Potter to get what he wanted—Harry was a freaking Horcrux, after all. And Delphini tries to flirt with Harry's son to get her way, telling him things like,
DELPHI: You're a positively disarming young man. (2.4.6)
Ooh, dang. That line is equal parts British and smooth.
Delphi toys with Albus because of a prophecy, but also because she sees Albus as easy to manipulate. And she has a point. Albus feels unloved by his own father, so it's easy for her to convince Albus that she cares about him. He doesn't know what real caring feels like.
But Scorpius isn't as easily manipulated. When Scorpius says Delphi shouldn't come with them into the past, she appeals to Albus. But even Albus realizes that Scorpius cares for him in a way Delphi doesn't, and he sides with his Best Bro Forever.
The rest of Delphi's plot unfurls by pure accident. When she learns that Albus and Scorpius accidentally changed the past to a place where Voldemort lived instead of Harry, she takes the Time-Turner and decides to try it again for herself. Ironically, she wants to do this by preventing Voldemort from killing Harry's parents.
That's right: she wants to stop him.
Harry can't let that happen, so he takes the shape of Voldemort to trick her. Delphi tells him,
DELPHI: I have devoted my life to being a child you could be proud of. (4.11.38)
You know that has to sting, because Harry's own son, Albus, would never say that to him.
Harry stops Delphi, who pleads,
DELPHI: I only wanted to know my father. (4.11.109)
(That right there might be the quote of the play, guys. You could put that into Harry's mouth, too. Or Albus'. All three of these characters desperately want to meet their dads—we're guessing that an alternate title for this play was Harry Potter: 2 Fast, 2 Freudian.)
Delphi doesn't get to meet her old man. As punishment, Harry keeps her from even seeing him.
But Delphi does get some parental proximity. After their fight, Delphi is sent to Azkaban where her mom, Bellatrix, is imprisoned. Maybe they can be cellmates?