Crystals can be used for the Synthesis of new items. Additionally, crystals can be traded to any NPC who offers Signet in return for Rank Points, which is sometimes needed to receive the next mission.
There are 8 kinds of crystals, with each element corresponding to a different crafting "task", and each can be found at the Auction House under the category "Crystals":
Fire Crystal - Burning or Heating
Pages in category "Crystals"
The following 26 pages are in this category, out of 26 total.