The Red Star
The red star is a one-of-a-kind star that is only found when you go back to the Gateway Galaxy (the first Galaxy you visit, where you meet Rosalina for the first time) via the portal on the platform after the Engine Room Dome. When you go back, climb to the top of the large structure in the middle to find Rosalina. After a short dialogue, a Red Luma will come and give you a Red Star, that transforms you into Flying Mario. You can now fly when you jump and spin. You must use this skill in this level to collect 100 Purple Coins. Once you do, the Luma will give you the Red Power Star. When you return to the Observatory, the Red Luma above the Library will reveal his secret to you, and allow you to use the Red Star's power in the Observatory. There's also one Red Star in the garage.
Once you beat the game, Purple Comets will enter orbit. If these are in orbit when you enter a galaxy, it will launch a challenge in which you need to collect 100 Purple Coins. You must collect them all to receive a Power Star. This allows you to find all 120 Stars, and unlock Luigi.