MIPS

MIPS is a rabbit that appears exclusively in Super Mario 64 within the basement after a certain amount of stars are collected. After fifteen stars, he will appear and run away from the player. If caught, he'll give the player a star. After fifty stars, he'll reappear again with the same gimmick, and give the player a second star.

Origin
The rabbit's name is derived from the MIPS microprocessor, Microprocessor without Interlocked Pipeline Stages, which is the instruction set used within the Nintendo 64. MIPS was actually one of the first characters modeled for Super Mario 64 hardware alongside Mario himself, where in an early prototype he would follow Mario around. He was kept due to a fondness by Nintendo staff.

In Super Mario 64 DS, MIPS does not return and is instead replaced by a group of rabbits. Internally, these rabbits are known as "MIP" and drop keys called "MIP Keys".

MIPS Throwing
In Super Mario 64's development, Mario was originally able to throw MIPS and the developers had the intent of letting Mario throw him by the ears. However, this wasn't done due to time constraints and perhaps some internal conflicts relating to animal abuse. Some players however swore by being able to throw MIPS, and vividly remember doing so. Many have then tied this to the Personalisation A.I. allowing MIPS to be thrown. With that, it's actually possible to throw MIPS in the most circulated version of the game but only in a body of water. This can only be done by clipping into the area where the pillars that control the moat level is found, and can only be done before the moat is drained.

Remnants of this function exist in the final game, such as an unused animation of MIPS bouncing on the floor as though recovering from being thrown. The animation can be viewed in game by changing the values. Code still exists for this animation being used when MIPS is thrown, although that's only possible in a body of water, which can be done via the aforementioned glitch.

MIPS Clip
MIPS Clip is the name given to a glitch commonly used in the 16 Star speedrun. The glitch allows Mario to clip through doors using MIPS, and can take MIPS with him if he precisely turns around by crawling and grabs MIPS, taking him through the door. This is used in a 16 Star speedrun to clip MIPS out of the basement corridors, and then to clip through the 30-star door leading to Dire, Dire Docks and Bowser in the Fire Sea.

Floating MIPS
Floating MIPS is an anomaly where MIPS will appear in their regular spawn location, though they will be levitating and have no textures on their face. They will also lack collision, making picking them up and obtaining their star(s) impossible. There is currently no known explanation for this anomaly.

Mipsy
Mipsy is a supposed "second MIPS" encountered in some personalized copies. They are often described as a pink MIPS that is much harder to catch, running faster and having a much more frustrating movement pattern. Like MIPS, they give the player a Power Star upon capture, but often only appear once. They most commonly appear in the third floor of Princess Peach's Castle, but they have been reported to appear in other places as well, most commonly anomalous locations. Whether or not they can appear alongside MIPS or if you have to catch MIPS to make Mipsy appear is inconsistent among reports. If they were planned by the developers, they were likely scrapped due to being too difficulty.