In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by Alexander Grothendieck (1966a), who named them crystals because in some sense they are "rigid" and "grow". In particular quasicoherent crystals over the crystalline site are analogous to quasicoherent modules over a scheme.
An isocrystal is a crystal up to isogeny. They are -adic analogues of -adic étale sheaves, introduced by Grothendieck (1966a) and Berthelot & Ogus (1983) (though the definition of isocrystal only appears in part II of this paper by Ogus (1984)). Convergent isocrystals are a variation of isocrystals that work better over non-perfect fields, and overconvergent isocrystals are another variation related to overconvergent cohomology theories.
A Dieudonné crystal is a crystal with Verschiebung and Frobenius maps. An F-crystal is a structure in semilinear algebra somewhat related to crystals.