Amethyst is a recent addition to the game of Minecraft (added in update 1.17) and allows the player to craft a few new items including a telescope, a brand new in-game item that allows a player to zoom in on distant objects without installing mods. The question is of course, where the hell do you find this new material, and once you have it, what exactly can you do with it?
find the amethyst
The only place in Minecraft where you will be able to locate amethyst is inside a geode set in the overworld dimension of your Minecraft world. These geodes will spawn in the underground portion of any biome, as well as in some oceans at level 70 or below (level 64 is sea level for reference). Of course, if you're playing in a old world, you will need to locate a new block generation area in order to access geodes and other new content in update 1.17, as they only appear in the world during the initial block generation.

One of the easiest ways to find a geode is to walk along a beach and look for a round structure of smooth basalt sticking out of the sand. The dark color of the basalt material contrasts sharply with the light-colored sand, making them easy to spot, even from a distance.
Alternatively, you can go swimming in the ocean and look for them on the ocean floor, or in hills and underwater ravines. They're a bit harder to spot this way, but the round, smooth structure tends to give them away, especially once the initial darkness of being underwater starts to wear off.

Once you find a geode
When you do manage to locate a geode, there's a good chance (95% in fact) that there's a crack in its exterior somewhere that you can enter through. If there is no crack or it is not exposed in the direction you are trying to enter, you will have to fight your way to reach the amethyst. Make sure you are using a pickaxe to ensure that what you mine falls as a block rather than just shattering. The outermost layer is Smooth Basalt, a new dark colored block that should add some extra detail to those moody builds.

Just inside the basalt is calcite, a new light-colored block that can only be found inside geodes and in bands in the Stony Spikes biome. As it stands, this block is purely decorative, but Minecraft is constantly evolving and you never know when blocks like this will have a new purpose. Also, since it doesn't have an infinite generation mechanic, calcite is a finite material in the game, making it a valuable commodity.

Finally, inside the calcite you will find your amethyst blocks and budding amethyst, with amethyst buds and amethyst clusters scattered over the various surfaces. It won't hurt you to mine all the Amethyst Blocks, Amethyst Buds, and Amethyst Clusters you find inside the Geode. Be sure to use a silk pick on the amethyst buds or they won't drop anything when broken.

However, if you want a renewable source of amethyst crystals to use for decorating or making stained glass, you'll want to leave the budding amethyst blocks untouched. Over time, these blocks will generate new Amethyst Crystals, growing them larger and larger until they reach their full potential, allowing you to grab them and craft them into blocks or other items.

Budding Amethyst unfortunately can't be moved in survival, not even with the Silk Touch enchantment or by pushing it with a plunger, so hopefully you can find a geode near where you planted your camp in your Minecraft world.
What to do with your amethyst
If you have access to a pickaxe with the Silky touch enchantment, you can extract amethyst buds (the first three stages of amethyst growth) from budding amethyst blocks to place as decoration. You can also mine amethyst clusters (the final growth stage of amethyst) for the same purpose. All of these will produce light when placed in the world (light levels 1, 2, 4, and 5 depending on the growth stage of the amethyst).

Alternatively, you can use any method to smash the amethyst clusters, causing them to drop amethyst shards, although a pickaxe is more effective and will drop 4 shards compared to just 2 for any other tool .
The Fortune enchantment can also be used to increase the number of amethyst shards dropped up to a maximum of 16 (although the average is around 8 with Fortune III). These shards can be used in three different crafting recipes.
amethyst blocks
The first recipe is the amethyst block. It is crafted by placing 4 amethyst shards into the crafting grid in a square pattern. Amethyst Block is currently only used for decoration, adding a cool crystalline look to your builds.

Tinted glass
Amethyst shards are also used to create stained glass, and they behave a little differently than stained glass. Stained Glass Blocks are crafted by placing a Glass Block in the center of the crafting grid and 4 Amethyst Shards around all sides (top, bottom, left, and right). Tinted glass blocks behave like normal glass to the player, letting light through so you can see. However, as far as the game engine is concerned, the blocks are opaque, preventing objects below from registering as being exposed to light from sources on the other side of the tinted glass.

The glasses
Crafted by placing a single Amethyst Shard in the upper center slot and 2 Copper Ingots in the other two slots in the same column (middle window and lower center slot), the spyglass allows the player to zoom in on objects and remote locations in vanilla Minecraft. This zoom effect slows movement and dramatically narrows the field of view when the spyglass is in use, so be careful not to use it if hostile mobs might be sneaking up on you.

That my friends is the story of amethyst. I hope now all this new information is crystal clear for you... I hope it's really calcify in your brain, allowing you to find and put to good use these amethyst geodes.