When it comes to crystal growing kits, they're basically a specific type of salt. In chemistry, a salt is a compound made up of positive ions (cations) and negative ions (anions). Cations and anions come together to form a crystal lattice structure. For example, table salt (sodium chloride) consists of sodium ions and chloride ions which will form cubic crystals. Many salts are soluble in water, which makes them ideal for crystal growth kits.
The basic creation process involves dissolving a dry salt substance in hot water to create a solution. As the solution cools, the dissolved particles come together and arrange themselves into crystals. This is usually pretty quick.
Each type of salt forms crystals with a unique structure—so some look like cubes, while others resemble diamonds. Various dry dyes and colorings are mixed in with the salts to give an array of different crystals colors. In nature these colors would typically come from iron, copper, chomium ions etc.