First up, what’s with the mouthful of titles? Let’s take them one by one: "dowager" means that her husband is dead; "viscountess" means that her husband was a viscount (higher up than the mere baronet like Sir Walter). Since "Dowager Viscountess Dalrymple" is quite the tongue twister, other characters address her as "Lady Dalrymple." You may also have noticed that Lady Dalrymple has a different last name from her daughter: that’s because Dalrymple is the name that goes with the title (so if she hadn’t had any sons and her cousin Jones inherited the title, he would still be the Viscount Dalrymple). Carteret, however, is the actual family name. So her names show that she ranks higher than the Elliots, or anyone else mentioned in the novel, and also that her rank is dependent on the men she’s related to.
Whew. That’s her name, what about her character? We don’t really see much of Lady Dalrymple in the book – her main purpose is to highlight how the snobbery of Sir Walter and Elizabeth turns into fawning when they’re faced with someone who outranks them. Lady Dalrymple's presence also shows how little Anne cares about rank when she turns down a last-minute invitation from the Dalrymples to keep an appointment with Mrs. Smith. Lady Dalrymple and her daughter also serve as an excuse for Lady Russell, Mr. Elliot, and Anne to talk about what makes for "good company" (16.16) and whether rank is at all relevant (Anne says no, Lady Russell and Mr. Elliot say yes). The Dowager Viscountess Dalrymple, despite her portentous name, is hardly a character at all, but functions more as a tool to reveal everybody else.