Quartz Crystal is a material you get in Mining Quest. When you find it you can give it to the new character (TBA) and they will join your party to fight enemies. Quartz Crystal can also be used in the Forge to learn the Charge move. It is one of the important items in the game.
Ruby is a material that can be found randomly. If you find ten rubies , ten sapphires and ten emeralds, you can buy a new Level 1 sword attack. The Attacks / Sword page will show more about this feature.
Sapphire is a material you can find randomly. It is simillar to Ruby. Sapphire allows you to pass the Sapphire Gate in the Maze of Mystery. Which requires twenty-five sapphires.
Emerald is much like Sapphire and Ruby. There is a Legendary Emerald in the opening cutscene, yet it doesn't exist in-game. Emerald allows you to pass the Emerald Gate in the Maze of Mystery. Which requires twenty-five emeralds. The Legendary Emerald can be hacked in with a code, if you play on an emulator.
Amethyst can only be found near granite. If you get it, you can use ten to unlock a new Level 2 sword special. It doesn't unlock a gate and doesn't do much except unlocking new sword specials.
Topaz is one of the rarest gems in the game. If you get it, you can use fifteen to unlock a new Level 3 sword attack. It allows you to pass the Gem Gate in the Maze of Mystery. You need twenty Topaz to unlock the Gem Gate. Topaz can be found in any layer, but it is most common in bedrock. (0.5% chance in Dirt and in Stone, 5% chance in Granite, 15% chance in Bedrock.)
Diamond is one of the rarest gems in the game. Like Topaz, it can be used to pass the Gem Gate and unlock a new Level 3 sword attack. You need only ten to pass the Gem Gate. Diamond can only be found in Bedrock.
Musgravite is the rarest gem in the game (that can be found). It can't be used to pass the Gem Gate but one is required to defeat Ttam as Star's Power is a special move that requires all gems. Only one of each though.
Legendary Emerald is the rarest gem in the game. It appears in the opening, but cannot be found in actual gameplay. It must be hacked in to be used and it has no use.