Which factors determine whether a substance will be amorphous or crystalline on solidification? What decides whether a substance will be crystalline or amorphous when it solidifies? 
I heard various folklores that the method of condensation of a liquid (fast or slow cooling) can be a factor which decides whether the resulting solid will be amorphous or crystalline. If this is true, is it possible that substances which are usually crystalline such as metals can end up being amorphous due to fast cooling? Seems no. Conversely, a substance such as glass which is usually amorphous can become crystalline due to slow enough cooling? Seems no. Then what decides?
 A: It is a fact and not folklore that transition to a crystalline solid or to a glass  strongly depends on the cooling rate. 
A slow cooling implies that the system stays  in the slightly supercooled  range of temperatures quite a long time. Long enough to allow  birth and catastrophic  growth of crystalline seeds (crystal nucleation). A fast cooling allows to get a range of temperatures small enough to  make nucleation processes very slow, while avoiding formation and growth of nuclei in the range of temperatures where they could grow faster.
Cooling rate is not the only parameter controlling glass formation. Also molecular structure or composition, for multi-component glasses, play an important role. all these ingredients contribute to the classification of a material as good or bad glass-former.
Just to provide some numbers, in the case of metallic glasses, cooling rates of about $10^4$K s$^{-1}$ are routinely used. Splat quenching technique allows about $10^6$K s$^{-1}$, while modern techniques may allow quenching rates as high as $10^{14}$K s$^{-1}$.
Notice that, even if it is possible to increase the quenching rate, thus making possible to get glasses in cases where it was not possible with slower protocols, there are cases, for example with polymeric materials, where even a very slow process cannot overcome steric constraints hampering crystal formation.
A: You are right, here are the details. 
In forming crystals in a metal after solidification from the melt, the material is seeking to minimize the energy invested in the bonds between adjacent atoms, giving rise to nanocrystals in the bulk which then grow at the expense of the higher-energy regions without long-range order. It is possible to "outrun" the kinetics of the crystallization process by quenching the metal from the melt so fast that nanocrystalline phases don't have enough time to form, giving rise to an amorphous solid galled a metallic glass instead of a crystalline metal. The quench rates required are heroic; on the order of ~millions of degrees per second. Metallic glasses are metastable; if subsequently heated to between 1/3 and 1/2 the melt temperature, they promptly crystallize.
It is also possible to get long-range order in ceramic glasses by cooling them extremely slowly, but note that the reason we get ceramic glass in the first place is the presence of substances in the glass mix which are added in order to interfere with crystal growth. These substances are called glass formers. 
Finally, note that it is also possible to promote the formation of a metallic glass by including alloy constituents in the melt which interfere with crystallization; these alloy metals are also called glass formers in this context. 
