Regina represents two important forces in this play: upward mobility and sex. She uses the latter to get the former.
Regina is the illegitimate daughter of Mrs. Alving's husband and her former maid, Johanna. As such, Regina is a daily reminder of Captain Alving's wayward life. She's young, vivacious, and attractive. (Don't be confused if Regina refers to Engstrand as "father": he married Johanna and "adopted" Regina to preserve their respectability.) None of the men in the play fails to comment on how she's "grown." Mrs. Alving knows it too, which is why she tries to protect Regina both from Engstrand – probably sensing his shady motives – and from her own son.
Regina doesn't mind using sex to get what she wants: security. She's already scheming to attract Oswald and almost lets it slip to Engstrand: "No; if things go as I want them to – Well there's no saying – there's no saying" (1.69). But we see early on that it's not about love, because just a few moments later she is asking Pastor Manders (indirectly of course) to consider her as a partner:
"Now, if it were in a thoroughly nice house, and with a real gentleman […] Then I should be glad to go to town. It's very lonely out here; you know yourself, sir, what it is to be alone in the world. And I can assure you I'm both quick and willing. Don't you know of any such place for me, sir?" (1.130-134)
Regina works all angles, and, like Engstrand, excels in giving people what they want. She's learning French for Oswald and acting pious for Pastor Manders. She would deny the association with Engstrand, though; "that filthy carpenter" is the last person with whom she wants to associate. When he asks her to come to town with him, she almost spits at him: "Me, that have been brought up by a lady like Mrs Alving! Me, that am treated almost as a daughter here! Is it me you want to go home with you? – to a house like yours? For shame!" (1.33).
Engstrand's brothel is the last place Regina wants to go, yet she's quick to recognize how things stand at the end of the play. She can't marry Oswald, so that door has been closed. Best to run after the other possibilities – Manders and Engstrand – who are leaving on the ferry. Mrs. Alving fears her demise, and Ibsen gives Regina a rare moment of emotional power. She looks sharply at Mrs. Alving and says, "I think you might have brought me up as a gentleman's daughter, ma'am; it would have suited me better" (3.150). And with that, she's on the steamer, most likely on her way to work at "Chamberlain Alving's Home."