From the Super Mario Wiki
Power Stars are collectible items that can grant Mario access to more levels in-game (like Flagpoles from the original Super Mario Bros. and the New Super Mario Bros. series); however, regular Super Stars only give limited invincibility. In Super Mario 64, Super Mario Galaxy, and Super Mario Galaxy 2 when the star for any given level has been collected, playing the level again will turn the Star into a translucent version of itself (similar to how Star Coins work). This Star can be collected, but it will not add to the player's total Star count. In all, there are 634 Power Stars in these three games combined. In Super Mario Sunshine, they are replaced by Shine Sprites.
Super Mario series
Super Mario 64 / Super Mario 64 DS
In Super Mario 64 and Super Mario 64 DS, Power Stars are spread throughout the worlds, and Mario (along with Yoshi, Luigi, and Wario in the Nintendo DS remake) is sent to go find them all. There are six Power Stars (seven in the DS remake) to collect in each world, and after finding 70 of them (or 80 in the DS remake), Bowser can be defeated. There are 120 Power Stars total in Super Mario 64; this was increased to 150 in the DS version due to 15 extra Secret Stars and 15 additional course stars. After a long quest, Mario manages to find all the Power Stars and restore them to Princess Peach's Castle, freeing Princess Peach and the Toads.
Mario Party 7
Super Mario Galaxy
There are four types of Power Stars: First, there are regular Power Stars which serve the same purpose in other 3D Mario games. The second kind is the three Green Power Stars that, in addition to acting as normal Power Stars, unlock the Green Launch Star in the Comet Observatory which leads to the Trial Galaxies. The third type is the Red Power Star. There is only one; it appears in the Super Mario Galaxy mission "Gateway's Purple Coins". Last are the seven Grand Stars, the first of which, and the only one obtained without a boss fight, is the first star in Gateway Galaxy. Subsequently, the other six Grand Stars are each collected by beating Bowser or Bowser Jr.. These power up the Comet Observatory and unlock new Domes to explore.
There are also Silver Stars, which aren't as powerful as the regular Power Stars. Five Silver Stars are equal to one Power Star. They can combine to form a Power Star.
Super Mario Galaxy 2
In Super Mario Galaxy 2, Power Stars play the same role as in Super Mario Galaxy. There are 242 Power Stars in total (116 normal Power Stars, 6 Grand Stars, and 120 Green Stars). If the Cosmic Spirit is used to complete a mission, the star collected will be a Bronze Star instead of a gold one.
Super Mario 3D World
While Power Stars themselves do not appear in Super Mario 3D World, the Green Star on the Goal Poles in the world map will turn gold when the course is completed with every character, and every Green Star is collected.
Mario Pinball Land
Stars looking like Power Stars reappears in the game Mario Pinball Land, working just like they do in Super Mario 64, obtained after completing certain trials, such as defeating a boss or a certain number of enemies in time, and their use is to make Mario's way to Bowser. At the end, the more the Power Stars obtained, the better the bonus in points.
Super Smash Bros. for the Nintendo 3DS / Wii U
Captain Toad: Treasure Tracker
Power Stars reappear in Captain Toad: Treasure Tracker, appearing at the end of each stage. However, the game does not count Power Stars, treating them more as Goal Poles in that they only serve to be the end of the level.
Names in other languages