Rachael has PTSD from being bullied. She likes certain anime because they offer some emotional solace and show great friendships.
A tsundere is a character who swings back and forth between being love-struck (dere-dere), and harshly critical (tsun-tsun) of their love interest. A tsundere is harsh, abrasive, and difficult to please. At first, anyway. Over time, she warms up to protagonist-kun and aw, isn't that sweet, she was only mean to him because she likes him! ^_^
The woman I love burns with jealousy,
leaps to conclusions, cries, and turns to ice.
But when she laughs, the world is mine.
— Yusaku Godai, Maison Ikkoku
Sub-types and Related Types
The "fiery redhead" may or may not be a tsundere, but has the bold, outspoken personality often seen in tsunderes. And there's a lot of overlap here, a lot of tsunderes have red hair.
TV Tropes breaks the 'tsundere' into two types. One is harsh and the other is sweet. This means their default or usual personality.
In the TV Tropes Analysis page, they also refer to classes of tsundere by their behavior and origin. The "wolf-girl" type is quick to violent outbursts. The 'discipline' type is orderly and in charge of things at their school, and berates their love interest for any perceived laxity on his part. The "dark past" type has well, a dark past, which is the explanation for her harsh tendencies.
A tsundere, similarly to a yandere, has an unhealthy reaction to feelings of love. But whereas a yandere reacts to intense, uncontrollable obsession with their love interest by becoming murderous and/or suicidal, the tsundere's violence level is usually non-lethal. She wants to hurt people sometimes, especially protagonist-kun, but with taunts, insults, and the occasional slap, punch or kick, never a deadly knife thrust.
A tsundere is sometimes a 'tomboy' or a tomboy with a hidden girly streak.
A related type is the 'himedere', a princess, literally or metaphorically, who has a haughty air (like Satsuki from Kill La Kill).
Similarly, the 'kamidere' type is a god, or a character who thinks they're a god (like Light from Death Note). Both of these types will have an aloof, "I'm better than you" attitude, but their personality may not come with a 'dere-dere' or sweet interior.
Some characters act like a tsundere, but it turns out they really don't love the protagonist, or they never really show a softer side. Asuka in Neon Genesis Evangelion is this type.
Most Well-Known Examples
Kyon from Fruits Basket and Inuyasha from Inuyasha are considered male tsunderes.
Asuka from Evangelion has all the "harsh" mannerisms, but hardly ever shows her "sweet" side. She's more of a deconstruction than a straight example.
Toradora's Taiga is the ultimate tsundere. She acts harsh and overreacts to perceived teasing because of her short stature, similarly to Edward in Fullmetal Alchemist. She puts up a tough front and projects a "don't mess with me" attitude so people won't pick on her for being short and/or flat-chested.
Other famous tsundere characters include Rin from Fate/Stay Night and Louise from Familiar of Zero.
Why is a Tsundere Appealing?
Most people's idea of a good day does not include someone yelling at them, putting them down, calling them an idiot, and maybe, if they're lucky, also an ass beating. But the appeal of the tsundere is not just that we're all a bunch of masochists on the inside. It has more to do with the fact that humans tend to appreciate anything more if it took more effort to obtain. In this case, winning the love or admiration of someone who hates you at first. It's also satisfying to audiences to see that an initially harsh character does have a softer side, making them more likeable, and it can be a form of character development.
Usually, a tsundere uses harshness as a mask or a defense mechanism to keep from being embarassed, picked on, or emotionally vulnerable. So the tsundere trope is about watching someone eventually grow in comfort and maturity enough to gain confidence and let their guard down more. Which is endearing and cute.
Tsunderes Outside of Anime
Tsundere-type women show up in a lot of romantic fiction. A girl being hard to get, even mean, is a turn-on for guys who like the "thrill of the chase" aspect of winning her over.
- Scarlett O'Hara acts like a tsundere in Gone With the Wind, to make a show of strength and play "hard to get".
- Estella in Great Expectations is kind of tsundere-like to Pip, but it's unclear whether she really has feelings for him, or anyone at all. Maybe she really is just emotionless, or maybe she's just being told to act that way by Miss Havisham. She overlaps with the kuudere, or "cold shoulder" type.
- Helga in Hey Arnold!. Some overlap with yandere (the creepy Stalker Shrine is yandere-like) and kuudere at times.
- Rainbow Dash in My Little Pony: Friendship is Magic acts like an anime tsundere at times, but not for romantic reasons. Discord is also kind of like this about ponies in general, and Fluttershy in particular.
- Leela in Futurama often hides tender feelings behind insults and kicks. (Fry deserves the insults though.)
- Miss Piggy from the Muppets is this, combined with the haughty attitude of a wannabe star.
- Cersei Lannister is like this with Jaime at times, and in general works really hard at maintaining a tough facade so others will be intimidated by her.
- Shakespeare brings us Kate from Taming of the Shrew, but he writes women this way a few other times as well.
The tsundere is a character type defined as going back and forth between harsh (tsun-tsun) and sweet (dere-dere) attitudes, especially with regards to their areas of sensitivity or someone they have a crush on. They are popular because it's a form of "playing hard to get", which creates the "thrill of the chase" aspect of romantic plot tension, and because audiences like seeing the "jerk with a heart of gold"'s heart of gold come out.
You can kind of think of the love of a tsundere as similar to the love of a cat vs. that of a dog. A dog loves you unconditionally. Which is nice, but somewhat boring, because it's less of a challenge. You have to earn a cat's interest, let alone affection. But that makes it all the more endearing and cute when the cat finally does warm up to you. So tsunderes are cats!