Power Gems are only seen in Crash Tag Team Racing.
They are purple gem stones that activate the gateways to each of Von Clutch's Motor World's lands. To obtain a Power Gem Crash needs to use Power Crystals to unlock the jump pads that lead to pedestals with Power Gems mounted on them. Upon reaching a power gem, a cutscene will play that shows how Crash takes it.
Power Gem Locations
Mystery Island Power Gem
The first power gem in the game is found on the Midway. After the second cutscene, Crash will ride an elevator down to the Midway, where straight outside there is a giant statue of Von Clutch. In his hand there is the power gem that unlocks Mystery Island.
Happily Ever Faster Power Gem
The second power gem in the game is found in Mystery Island. It is up at the top of the area, around where Cortex is standing. To unlock it, it costs 8 power crystals.
Tyrannosaurus Wrecks Power Gem
The third power gem in the game is found in Happily Ever Faster. It is at the top of the area to the right of where the race gate for Evilocity can be found. To unlock it it costs 10 power crystals.
Tomb Town Power Gem
The fourth power gem in the game is found in Tyrannosaurus Wrecks. It is through an alcove with lava in it.
Astro Land Power Gem
The fifth power gem in the game is found in Tomb Town.