There are three types of gems to find in the game: Rare Gems (20), Sparkly Gems (10), Shiny Gems (8) for a total of 38 gems to collect. They can be traded in at the basement shop outside of the Stoc n Barrel early in the game. These are used to purchase Gem Accessories, upgrade those accessories, and unlock Allcat Doors. Collecting all of the gems and using them for the tasks listed will unlock a bunch of achievements / trophies. None of them are missable as you can back track from the final autosave before completing the game.
Timestamps:
0:00 Rare Gem #1 (Road to Ikenfell)
0:22 Rare Gem #2 (Ikenfell Grounds)
0:37 Rare Gem #3 (Southern Dorms)
1:11 Rare Gem #4 (Southern Dorms)
1:42 Rare Gem #5 (Southern Dorms)
2:12 Rare Gem #6 (Alchemy Lab)
2:35 Rare Gem #7 (Alchemy Lab)
2:55 Rare Gem #8 (Forest Hidden Path)
3:14 Sparkly Gem #1 (Northern Dorms)
3:38 Rare Gem #9 (Northern Dorms / Gilda Boss Fight)
3:57 Rare Gem #10 (Snatcher's Lair)
4:11 Rare Gem #11 (Twilight Yard)
4:51 Rare Gem #12 (Twilight Yard)
5:15 Rare Gem #13 (Astronomy Tower)
5:28 Rare Gem #14 (Great Hall)
5:44 Rare Gem #15 (Great Hall)
6:11 Rare Gem #16 (Spirit Oval)
6:39 Rare Gem #17 (Spirit Oval)
7:08 Sparkly Gem #2 (Great Hall)
7:26 Shiny Gem #1 (Cemetery)
7:42 Sparkly Gem #3 (The Roost)
8:07 Shiny Gem #2 (The Roost)
8:25 Sparkly Gem #4 (Forest Ruins)
8:48 Rare Gem #18 (Dueling Hall)
9:01 Shiny Gem #3 (Dueling Hall)
9:10 Shiny Gem #4 (The Archives)
9:33 Shiny Gem #5 (The Archives)
9:48 Sparkly Gem #5 (Dueling Hall)
10:10 Shiny Gem #6 (Gardens)
10:21 Shiny Gem #7 (Stacks)
10:31 Sparkly Gem #6 (Stacks)
11:06 Sparkly Gem #7 (Ikenfell Grounds)
11:21 Shiny Gem #8 (Spelltower)
11:44 Sparkly Gem #8 (Spelltower)
11:57 Rare Gem #19 (Orchard)
12:18 Sparkly Gem #9 (Ghostly Grove)
12:44 Sparkly Gem #10 (Dueling Hall / Allcat Lair)
13:35 Rare Gem #20 (Ikenfell Grounds)