The basic idea of a time crystal is relatively straight-forward. A crystalline medium has a periodic, or regularly repeating structure. However, because of entropic considerations (forcing the substance into its lowest energy state) the crystal will not have the same repeating structure in all directions: it will be asymmetric -- this is known as symmetry breaking of spatial translation symmetry. So whereas with normal crystals this repeating, periodic structure is asymmetric spatially (the spatial configuration of the crystalline lattice); with a time crystal the asymmetric periodicity is not in spatial organization but in time-varying media.
Ultracold matter normally serves as the medium, where ions are cooled to such a low temperature that there is no longer thermal noise and they are confined within a magneto-optical trap. This should be a completely stationary arrangement, however because of a time varying oscillation driven into the ions they will interact with quantum vacuum fluctuations and the medium will rotate. This is a similar and related phenomenon to the dynamical casimir effect, where photons and phonons are emitted from the vacuum due to time-varying boundary conditions. In the case of the time crystal, this would allow for a perpetually rotating device.
The reason this had not been observed until recently is that quantum processes are not governed by time-dependent variables, mainly because quantum particles at equilibrium do not have definite positions. However, Ytterbium ions in a magneto-optical trap can be made to undergo definite localization, and driven into non-equilibrium when subject to a periodic drive -- in this case a laser that periodically flips the spin of the ions; all of which causes them to become time-correlated. Now, since there should be a time-symmetrical periodic oscillation of the ions (an emergent sub-harmonic frequency of the rate of flipping the spin with the laser); time-symmetry can be broken, and a discrete time crystal is formed. The ring of trapped ions begins to rotate.
It is extremely interesting that such a situation is driven by the quantum vacuum fluctuations -- like the dynamical casimir effect, the observation of an actual time-crystal is a clear demonstration of the real energy of the quantum vacuum.