A time crystal (or poH qut in Klingonese) was a rare mineral with a non-equilibrium matter state. Time crystals could be identified by their orthogonal indices. To Klingons, at least those outside of Boreth, time crystals were a myth, a symbol of Kahless, and the namesake of Qo'noS. (DIS: "Magic to Make the Sanest Man Go Mad", "Through the Valley of Shadows")
By the mid-23rd century, the Vulcan Academy taught its students about time crystals. No Federation-aligned species had been able to stabilize them as the decay rate of the lattice was too unpredictable, and it was thought that any technology based on time crystals must have been perfected by a four-dimensional race. (DIS: "Magic to Make the Sanest Man Go Mad") Nonetheless, time crystals were encountered a number of times.
Time crystals were a rare mineral native to the planet Boreth, and the monastery on the planet was not just built to house the Followers of Kahless, but also to protect the mineral. Guarded by the Timekeepers for generations, the crystals generally did not leave the walls of the monastery. The Klingons did at one point exploit the crystals, but stopped because the power to manipulate time was a weapon unlike any other. (DIS: "Through the Valley of Shadows")
Project Daedalus crystal
A modified time crystal was required to enable the Red Angel time travel suit developed by Section 31 in the 2230s. Based on intelligence from an operative on Qo'noS, Leland learned that one had hit the black market near the Orion outpost and acquired it. On Doctari Alpha, the crystal was given to Gabrielle Burnham, who planned to charge it from the nearby supernova of Alpha Lupi. However, the Klingons tracked the crystal to the colony and attacked, forcing Burnham to install the crystal in the suit prematurely. Burnham used a device which attracted the crystal to install it in the back of the suit rather then using her hands. After Burnham was trapped on Essof IV in 2257, the time crystal in her suit was targeted by Control within Leland's body, causing an explosion. (DIS: "The Red Angel", "Perpetual Infinity", "Through the Valley of Shadows")
Harry Mudd's crystal
At one point, Harcourt Fenton Mudd acquired a time crystal device capable of creating chronal displacements. This technology consisted of two components: a wrist-mounted crystal with controls in an unknown script, and a much larger one on his ship capable of generating the required energy for chronal displacements.
It was suspected that he used this device to rob a Betazoid bank, and in 2256, during the Federation-Klingon War, he used it to attempt to steal the USS Discovery and sell it to the Klingons. The device on Mudd's wrist disintegrated after he allowed the Discovery to rejoin the time stream. (DIS: "Magic to Make the Sanest Man Go Mad")
Second Red Angel suit
Following the appearance of the fourth Red burst at Boreth in 2257, Captain Christopher Pike of the USS Discovery went to the planet's Monastery. At first, the Timekeeper Tenavik refused to give Pike a time crystal, despite Chancellor L'Rell's permission. However, Tenavik allowed Pike to take one. As they ventured to where the time crystal were kept, Tenavik told Pike of their power capable of advance aging of plants and Tenavik himself. As Pike attempted to take a crystal, Tenavik warned him that he would see his future and his fate would be sealed. Despite his vision, Pike was able to take one back to Discovery. (DIS: "Through the Valley of Shadows")
When Discovery failed to auto-destruct, Burnham and the crew decided to use the time crystal to take Discovery's and the sphere data to the future. When a fifth red burst appeared, Discovery took the crystal to Xahea. There, Xahea's queen, Me Hani Ika Hali Ka Po was able to use her Dilithium incubator to power the crystal. The crew was able to rebuild the suit and install the crystal, despite Burnham receiving a vision of a future where control kills everyone. During the Battle, Burnham was able to use the crystal to create the first five red bursts. After creating the sixth red burst, Burnham, Discovery and it's crew were able to travel to the future through a wormhole. However, Burnham landed on Hima in 3188. There, she sent the suit and the last red burst through the wormhole. The crystal along with the suit self-destructed and were destroyed. Discovery landed on The Colony on 3189. (DIS: "Such Sweet Sorrow", "Such Sweet Sorrow, Part 2", "That Hope Is You, Part 1", "Far From Home", "Die Trying")