There is a certain amount of sugar that will dissolve in any volume of water. This solubility changes with temperature, the hotter the water, the more sugar will dissolve. This means you can heat water to dissolve a lot of sugar, then cool it, and some of the dissolved sugar will crystallize out of the water. In order to grow beautiful, big, ordered crystals this process must be slow! One way to do this is to let the water slowly evaporate, effectively increasing the concentration of sugar slowly over time. The type of sugar should not matter if it is fully dissolved, but if it is not heated high enough, it may very well affect the crystallization process. Different types of sugars (sucrose vs fructose vs lactose) will likely crystallize differently (although I am not sure what sugar is best). Typically crystallizations work best with highly pure materials, so brown sugar likely wouldn't crystallize as well as highly refined (purified) white sugar. The crystal growth of a compound like sugar depends both on the different elements and the overall shape of a molecule. It is difficult to predict how differences will affect crystal growth.

Any pure, solid, substance, will form crystals if oversaturated in a solution of water. If you dissolve egg white in water and then let it dry up, it will form crystals (these crystals may be too small to see, but they will be there). Crystals are just the most efficient way for atoms to fit together in a solid, so solids naturally crystallize, given the opportunity.