Crystals are an endgame content system that can be obtained through grinding or donating.
Christian/xerxes is the primary developer for the Crystal system.
There are 4 types of crystals right now:
Unknown death is vastly different from the other two and is priced much higher.
Warlord is the next expensive with Eversince tending to be the cheapest.
Decades came in a later update, and are the first items to be "stackable". Read more on Stacking.
As stated above crystals can be obtained through grinding or donating - to grind/farm you need at bare minimum a gun with the talent Crystal Cavern.
Each Crystal Cavern gives you a 1/4000 (1/2718 if you are Player Level 200 or higher) chance to obtain a crystal each time you ID a body.
To better your chances you want to be at least player level 200, have the power-up Chrysalis of Eternity, and have multiple Crystal Caverns on your weapon.
People before us have done the math on your chances with multiple Crystal Caverns and/or Chrysalis and posted them in a google sheet.
To get multiple Crystal Caverns, there are a few methods.
Unknown Deaths (UDs) are the most sought-after crystal.
Like all other crystals, they are usables, but unlike the others they can only be used on guns of Cosmic or Planetary rarity (not extinct or any other).
They destroy the gun you use them on irreversably, but have a chance to do the following:
If the Unknown Death is used on a Unique gun, effect #5 cannot happen.
In addition, the resulting Gun Mutator cannot be used on a unique gun.
According to Christian, the official chances for UD are 5/9 for effect 1, then 2-4 are equal chance, and 5 is about 10x less likely than those three.
The percents above are an estimate based off live values provided by the in-game display. The raw numbers are available here:
There is also a list of talents that are either not on a Cosmic+ prefix or are blacklisted and thus cannot be obtained with #4:
This is not a complete list.
Warlords are the next most sought after. Like other crystals they are usables that can be used on weapons.
Warlords in particular modify the talents of a weapon. Their effects are as follows:
Eversinces are usually the cheapest. Like other crystals they are usables that can be used on weapons.
Eversinces in particular modify the stat values of a weapon. Their effects are as follows:
These are fundamentally different from other crystals. Firstly, they are account/trade locked.
Second, they aren't usable on guns, but instead can be stacked and redeemed in batches of 10 for either a Crystal of the Warlord or Eversince. (50/50 chance)
Finally, Decades are earned differently from the other 3 crystals. You get them from the talent Fungal Cavern, which has a flat 1/350 chance to give you one Decades after each kill.
The ascended power-up Radiant Heart doubles the proc effect to give you two Decades per talent, per proc.