Princess Toadstool, or Peach as she is known nowadays, is possibly Nintendo's most prominent female character in their line-up. She has been in the most games out of any Nintendo woman by far, and I'd have to imagine she's the one that leaps to mind for most people when they think of a female Nintendo character. In her very first appearance in the seminal Super Mario Bros., she is relegated to the very last moments of the game as the damsel in distress trapped by the villainous Koopa King Bowser.
Character Design and Personality
Princess Toadstool is curiously shorter than Mario in this game (that would change in the US Super Mario Bros. 2), and her sprite does not match the concept art for her at all; she has red hair, a white and red dress, and tanned skin. This color scheme is unique to this one appearance, as she adopted the pink dress and blond hair (and a fairer complexion) in Super Mario Bros. 2 (USA) and would stick with that for the rest of her appearances. Even the Japanese boxart includes her more conventional look:
To be honest, Pauline had more personality in terms of debuts. Peach has no animation, and after this brief message, she vanishes from view. Not the most becoming of introductions for such an iconic character, I'd say.
Impact on Narrative
Here, Peach is the definition of the “damsel in distress” trope. The point for playing the game is rescuing her. So... she is the narrative force, which is not a great motive in theory.
She has become one of Nintendo's female stalwarts. She became playable in several Mario titles (mostly in outliers like the Mario Kart series, Mario Party, and the RPGs, although Super Mario 3D World is an excellent return to her being playable in the main Mario platforming line), and was the second woman added to Super Smash Bros. (behind Metroid's Samus Aran). She has, when given the opportunity (typically in the Paper Mario series), a fairly strong-willed personality that shows that she can be quite capable despite her imprisonment. She is often featured in Super Mario merchandise, making her the most marketed woman in Nintendo's stables.
Peach (along with Pauline and Princess Zelda) helped establish the long-running, and frankly repugnant, industry standard for women as gaming characters. Peach continues to be featured as a damsel in most of the key Mario platformers, limiting the positive agency mentioned above. Outside of the Paper Mario series, Peach is very much a stereotypical princess in the Disney mold, with a high-pitched voice and a vapid demeanor towards the action going on around her. She embodies feminine tropes (liking pink, using hearts in her attacks in certain games, utilizing her emotions to solve puzzles in Super Princess Peach, etc.).
Peach has never been a favorite character of mine. She's so entrenched into the damsel/princess ideal that her characterization rubs me the wrong way most of the time. Super Mario Sunshine is probably the biggest offender in marking Peach as an insipid character to me. However, I do support her Paper Mario appearances, as Intelligent Systems added some awesome depth into her personality, and gave Peach some much needed agency. On the whole, though, I'm not her biggest fan. Her flaws stick out more to me than her strengths.
Listing Peach's future appearances would be a mammoth task, so I'll touch on the three that I think made the biggest improvements to her. Super Mario Bros. 2 (USA), where Peach became playable for the first time, is #1. Her abilities were among the best in the game, particularly her floating jump, and made her a viable (and at times vital) choice for progression. The second is Super Mario RPG, which gives Peach a chance to serve as a character in Mario's three character party. She fills the role of healer, an unfortunate trope of women in RPGs, but at the very least she served a key component of many of the possible teams in the game. It gave her some characterization outside of being a damsel as well, which Intelligent Systems would expand upon in their RPGs to come. Lastly, I would argue that her addition to Super Smash Bros. Melee was her grandest hour. Being able to fight on even ground with many of Nintendo's other iconic heroes and villains was a huge increase to her agency, and she has been considered one of the better fighters in the series.