MIPS is the name of the yellow rabbit that appears in the basement of Peach’s castle. In order to access the basement, you will need to clear Bowser in the Dark World. This will net you the key to the locked door through either wooden door in the main lobby, at the bottom of the stairs. Down here, head through the next normal wooden door to find an area with greenish-colored walls. MIPS will be located in the middle of this area once you have at least 15 Power Stars in your possession.
When you approach him, MIPS will run away and it is your goal to chase after him. The best thing to do is try and bait him into running to a dead end, as you can dive towards the middle of the pathway when he’s trying to escape to grab him. Upon finally catching MIPS, he will hand over a Power Star. Note that he will appear a second time in the basement, but only after acquiring 50 Power Stars. There’s really nothing different the second time around, as you need to do the same thing as before to get a second star. Once you get this second star from MIPS, he will not appear again.