Let's start with clarifying the terms 'Crystal" and "Growth".  The word "crystal" originates from the Greek κριος (coldness) or κριµος (ice).  The father of crystal fabrication technology is A. Verneuil with his flame-fusion growth method he described in 1902. 

What are crystals?  Crystals are ordered arrangements of atoms (or molecules).  Materials in crystalline form has special optical and electrical properties, in many cases improved properties over randomly arranged  materials(also said to be amorphous or glassy)

What causes crystals to "grow"? The driving force for crystallization comes from the lowering of the potential energy of the atoms or molecules when they form bonds to each other. 

The crystal growth process starts with the nucleation stage.  Several atoms or molecules in a supersaturated vapor or liquid start forming clusters; the bulk free energy of the cluster is less than that of the vapor or liquid. The total free energy of the cluster is increased by the surface energy (surface tension), however, this is significant only when the cluster is small. A cluster of radius smaller than a critical radius, r^*will evaporate (or dissolve in the solution) a cluster of radius greater than r^* will become stable, will increase its size by the addition of other atoms and is thus "growing"! The critical radius r^* also defines a critical energy barrier, DG, that we need to overcome in order to obtain a stable nucleus that will keep growing, eventually become a large single crystal!

Thermodynamics can help us describe the process. Assuming a spherical shape for the nucleus the free energy of its formation is:

DG = 4P r² s + (4/3)P r³ )G_v

where DG is the total free energy; r is the radius of cluster; s is the surface tension; DG_vis the free energy change per unit volume forming the stable solidification from vapor or liquid. The total free energy DG goes through a maximum DG^*at a critical radius r^* which can be obtained by derivation of total free energy as given above with respect to radius and solving:

(dDG^o/dr) = 0

Methods

 Two frequently used crystal growth methods are Physical Vapor Transport (PVT) and Bridgman methods. 

PVT method is the crystal growth under vapor - solid equilibrium conditions. The temperature of the starting material (powder form)  is higher than the nucleation/crystal growth region.  This imposed temperature gradient leads to a mass flow resulting in a net mass transport of vapor species towards the crystal growth site. The vapor species may consist of molecules of the material itself, such as PbI₂(solid) ® PbI₂(vapor), or dissociated into its separate constituents, such as CdTe (solid) ® Cd(vapor) + ½ Te₂(vapor), and residual gases.  The reverse process occurs when vapor species nucleate and then continue to condense on the crystal growth interface at a rate of 3-5 mm/day. A typical PVT grown vanadium doped CdSSe single crystal is shown below currently being investigated for its optical properties. 

The Bridgman growth method is basically a controlled freezing process taking place under liquid - solid equilibrium conditions. The growth also takes place under a temperature gradient, and the mechanism is to produce a single nucleus from which a single crystal will propagate and grow.  This is achieved by allowing the solid - liquid interface to move slowly (5-50 mm/day) until the whole molten charge is solidified.  A PbI₂ single crystal is shown in the figure below. 

Compared to other growth methods, Bridgman method is considered to be a rather simple crystal growth method, but several limitations still exist.  The Bridgman method can not be applied to a material system which decomposes before it melts, systems having components with high vapor pressure, and materials exhibiting destructive solid - solid phase transformations which will compromise the crystalline quality on cooling the crystal at the end of the growth run.  PbI_2, lacking such a phase transition  can also be grown by Bridgman method.