Peach is a character in Nintendo's popular Mario series. Apparently we are to assume that Princess Toadstool, is in fact this character, whose name first appeared in Super Mario 64 for the N64, and whose name persisted through the continuations of the Mario saga, as seen in Super Smash Bros. 2 (Melee).
On what premise may we surmise that Peach is in fact Princess Toadstool? In Melee, Peach wields a frying pan as one of her attacks. In Super Mario RPG for the SNES, after completing a series of sidequests, Princess Toadstool is able to attain her most powerful weapon, the frying pan. Thus, it is implied that they are the same person.
Throughout most of her video game career, Peach has played the stereotypical female role of, "Oh no! I got caught by the bad guy! Protagonist, oh protagonist, save me!!!" A clash with the norm appeared in Super Mario Bros. 2, when Peach was a playable character whose special technique was temporary hovering. In Super Mario Bros. 3, Peach was kidnapped by Bowser, the same in Super Mario World, the same in Super Mario 64. As mentioned before, Peach's first GameCube debut appears as a playable character in Super Smash Bros. Melee, using her loyal retainer Toad as a shield while beating people with her parasol and frying pan.
Note: The origin of the name Peach is the fact that in all of the Japanese versions of the Mario series, the Princess' name has been Peach! But this was not revealed to the American populous until much later in the series.