Satellite Love Interest: He has slightly more screen time than The Prince in Snow White, but ultimately he represents more of a goal than a person—there were deleted scenes that gave him more of a personality, but they were cut from the film. Disney noticed this, and made up for it in the third movie.
Abusive Parent: Not only with her stepdaughter (physical and mental abuse by the wazoo) but her own biological daughters as well (is very psychologically abusive to Anastasia in the sequels, and she's the reason why Drizella is an asshole as well. Also, Drizella is absolutely terrified of her.)
Dangerously Genre Savvy: She manipulates her daughters into tearing Cinderella's dress apart, and discreetly trips the footman to break the glass slipper.
She's also this in the third movie. Housekeeping comes to your room before you ring the bell? It's your stepdaughter trying to sneak in.
In Twice Charmed, she realizes that Cinderella might be recognized by the Prince, so she tells Franco to shrink her. Then, she has him use his magic to make Anastasia and Drizella graceful and beautiful so they'll catch the Prince's eye.
Also implied to be the reason she married Cinderella's father (and indeed, Drizella and Anastasia's). And since she was widowed twice, while the girls are still young, this implies more...
Hate Sink: She is considered one of the most despised Disney villains.
I Gave My Word: However, she is very specific about what words she uses exactly. When she says that Cinderella can go to the ball, she very clearly states if she can finish all her chores and if she can find something suitable to wear. When she surprises Tremaine by revealing she has finished her work and has a dress, rather than go back on her word, all she has to do is point out the beads Cinderella is wearing to get her daughters into a manic frenzy when they think Cinderella stole from them.
Drizella: Mother do you realize what you just said?!
Lady Tremaine: Of course. I said "if."
It's All About Me: A textbook example; all she cares about is getting the status she believes that she deserves.
Lazy Bum: Essentially; she forces her own stepdaughter to do all of the work for her, some of which is work that she could either easily do herself or at least just pay somebody to do (she's already filthy rich so it's not like paying a full staff would harm her fortune in any way) and makes her daughters marry into money just so she can bump her own status without having to do any work herself.
This is more out of an attempt to destroy Cinderella's beauty more than anything.
And according to the opening narration, the money is squandered on the stepsisters, leaving the estate in financial trouble.
Manipulative Bitch: Manipulates Anastasia and Drizella into destroying Cinderella's dress.
The Sociopath: Look at the signs. Domineering, manipulative, abusive, has no problem lying, derives pleasure from hurting/humiliating others, has a grandiose sense of what she's entitled to.. Along with her general antisocial behavior, she may be a textbook example of this trope.
Easily Forgiven: Cinderella holds no ill will towards Anastasia for all her antagonism, actually helping her to get together with the baker. To be fair, Anastasia had shown quite a bit of Hidden Depths by that time.
In the original Cinderella fairy tale, it was mentioned that one of the stepsisters was actually nicer to Cinderella than the other, if only somewhat. Also, in most retellings of Cinderella, one of the stepsisters, usually the younger stepsister, becomes nicer to Cinderella.
Hidden Depths: Anastasia is revealed to have it in the third movie. In example she's more naive than people would think, actually longs to find love for herself, and is a female "Well Done, Son" Guy for her mother.
Ineffectual Sympathetic Villain: Anastasia becomes this in the sequels when it becomes clear that, with Cinderella out of the picture, she is now the lowest ranked in the family's pecking order.
Woman Child: Anastasia comes off as one in the third movie.
Yes-Man: Drizella in the sequels shows more shades of this towards her mother, though it later shows that this not just because Drizella is spoiled rotten and despicable in her own right, but because Tremaine threatens her.
Cat Stereotype: Lucifer is dark grey and grey with a black head and off-white muzzle, and fits both black and grey cat stereotypes by being evil, fat and lazy.
Determinator; He'll stop at nothing to catch the mice. Even getting stuck in Drizella's dress (in the first movie) and being turned into a jack-in-the-box (in Cinderella III) doesn't stop him. Nor does becoming a human.