Character Analysis
Ivy is the master of her own domain. When Hilda and Connie interview her, they see her as mostly attentive and polite:
"[She] seemed quite nice, spoke with a bit of a broad slur, but in heavily correct English, and from having bossed the sick colliers for a good many years, had a very good opinion of herself, and a fair amount of assurance. In short, in her tiny way, one of the governing class in the village, very much respected." (7.107-08)
What we see here is that Tevershall has its own hierarchy, and Mrs. Bolton is at the top. As the lady of Wragby, it should be Connie who is respected—but Connie lacks Mrs. Bolton assurance and self-satisfaction.
Mrs. Bolton's story is just the kind that politicians like. An ordinary housewife with two children, she was left all on her own after her husband died in the mine. As she tells it, the mine owners really screwed her over. They insisted that the accident was her husband's fault and so only gave her a little bit of money, and they wouldn't even let her take it all at once. She had to go stand in line for hours once a week to draw her measly thirty shillings—just enough to keep the family going for a week. But she's got a lot of determination, Ivy does, and she managed to get a nursing certification and then a job working for the parish—the church government—doing the nursing for charity cases. Rags to riches: you have to admire it.
Living Up to Her Name
Once Ivy joins the staff at Wragby, we can see why she's got the name "Ivy." She's a clinger. She gets her tendrils around Clifford and gradually starts to squeeze the life out of him. Well, not literally, and not life so much as, for lack of a better word, class: "Gradually, with infinite softness, almost with love, she was getting him by the throat, and he was yielding to her" (9.17). Mrs. Bolton knows how to manage men. Unlike Connie, who is too direct. If she doesn't like something that Clifford does, she just up and says it.
Mrs. Bolton is way more subtle. She thinks of men as big babies: "Why, I've handled some of the toughest customers as ever went down Tevershall pit. But let anything ail them […] and they're babies, just big babies" (9.18). And this works for her. It's especially effective with Clifford, who likes the way that she idolizes him. The lady has a thing for the aristocracy. She lets Clifford teach her chess—just well enough so that she never quite beats him—and Connie rolls her eyes at Mrs. Bolton getting all "flushed and tremulous like a little girl, touching her queen or her knight with uncertain fingers" (9.31).
Common
The thing is that Ivy Bolton is common. She doesn't have the natural breeding that Mellors does, even though they come from basically the same background. We know this because she's obsessed with money. When she and Connie find a traveling case in the attic, Connie sees it as soul-crushingly hideous, while Mrs. Bolton thinks it's beautiful: "what beautiful brushes, so expensive, even the shaving brushes, three perfect ones! No! and those scissors! They're the best that money could buy. Oh, I call it lovely!" (11.18). Things are beautiful when they're expensive. She's got no aesthetic sense at all and, this passage suggests, no soul.
We can also tell that Ivy is common because she's obsessed with village gossip, and she relates it to Clifford in a "stream of gossip […] better than any book" (9.40). Clifford loves it. It's just like watching trashy TV: he can't take his eyes way.
To give him credit, he does know that it's not good for him. "When I hear Mrs Bolton talk," he says, "I feel myself plunging down, down, to the depths where the fish of human secrets wriggle and swim […] horribly, among the sea-weeds and the pallid monsters of the very bottom" (17.98). But it makes him common just the same, bringing him out in "his true colours: a little vulgar, a little common, and uninspired; rather fat" (9.36).
The way that Ivy makes Clifford common helps support Lawrence's idea that you're just born with class—or not. Clifford thinks that you can educate anyone into any station of life, but Lady Chatterley's Lover disagrees. According to anyone with common sense, Mellors and Mrs. Bolton should be a pair: they're both commoners, from the same village; they both work with their hands. It's a match made in heaven.
But actually it's not. It turns out that Mrs. Bolton is a better match for Clifford, and Mellors is a better match for Connie. Yeah, there are common and uncommon people, but it has nothing to do with any societal structures.
Commonality
But there's also a deeper connection between Connie and Mrs. Bolton, one that transcends class because it's based on the "natural" commonality of gender. They're both women, and that makes them the same. They both recognize it. Connie sees that Mrs. Bolton is "somehow, in her femaleness, an ally" (16.23), and they work together in the garden talking about husbands and men just like any sitcom wives.
Shared femaleness also makes Ivy half-appreciate Connie's desire to leave Clifford, even though Ivy really sympathizes more with him: "I'll be faithful to Sir Clifford, and I'll be faithful to you, for I can see you're both right in your own ways" (19.157). Lines like this make it hard to tell what exactly we're supposed to think about Ivy. From Connie's perspective, she's vulgar, common, and managing; but she does help Clifford start to care about his mines and his workers again.
It's tough to say what her angle is. She can't possibly expect to marry Clifford, and it seems like all she wants to get out of the relationship is proximity to an aristocrat. Is that her fault? Or is it the fault of an unnaturally class-based society?