Crystals are a form of miscellaneous loot. They are often found whilst mining or through random events.
Their main use is in the process of crafting, boosting a certain attribute of the product, depending on crystal type. Note that only up to three crystals of the same color can be used. For further details, check the CE Data Repository (CEDR) You can find them under the "Misc" category.
Their other use is in certain NPC missions.
Please note that, whilst crystals sound and look like they may be valuable, they may not be worth as much as you originally thought on the market.