Princess Peach's Castle, also known as Mushroom Castle or Castle Toadstool, is the home of Princess Peach Toadstool and the centerpiece of Toad Town. It appears in a plethora of games; Most notably in Super Mario 64 and it's remake, where it is the Hub of all Worlds.
Throughout the years the Castle has varied in appearance greatly, it is still quite likely caused by variety in artistic direction, and not actual redesigns of the Castle. The following list describes the appearance of Princess Peach's Castle, game by game, in the order of which each game was released.
In Super Mario 64, the Castle and it's surrounding areas make up the Hub, providing access to the game's Worlds. It is in this game that we are given our first, in-depth look at the Castle, establishing many of its recurring themes; Including it's stain-glass Portrait of the Princess, it's bridge & moat, red-tile roof, and it's light grey brick exterior.
In Super Mario Galaxy, Bowser uses the power of the Grand Stars to transport the Castle, with Princess Peach within, to the Center of the Universe. In appearance, the Castle looks like a modernized version it's Super Mario 64 design, maintaining all major elements from its Predecessor.
In Paper Mario, the Castle is nearly identical to its design in Super Mario 64, differences include the stain-glass window being replaced by a bronze Super Mushroom Emblem, the Castle's flags changing in color from red to blue, and some of the spare rooms being put to use; For example, there is now a door at the top of the stairs in the Main Hall that leads into the Ball Room, which in turn connects Peach's Room, and the Castle Balcony.
It only appears briefly at the beginning of the game, when Mario and Peach are taking a walk, Lakithunder attacks the Castle, distracting Mario long enough for Bowser Jr. to kidnap the Princess. It looks aesthetically identical to its appearance in Super Mario 64.
It has appeared multiple times throughout the Mario Kart series, beginning with Royal Raceway in Mario Kart 64. It then appears in Mario Kart: Double Dash!! in Mario Circuit and is located in the center of the course. It later appears in Mario Kart Wii in Mario Circuit once again. Later in Mario Kart 7 for the Nintendo 3DS, Mario has his own track based on Peach's Castle. Here, you can actually drive IN the castle in a loop before gliding back to the ground.
The Castle is also used as a Stage in Super Smash Bros. Melee, and has a bit more of a brownish brick color to it rather than the more white color that has been seen in other Mario games. Players fight on the roof of the castle in this game. The castle also appears as an unlockable trophy throughout the game.