Why do elements light up into colored flames when set on fire? And why are they different colors?

The colored light seen from burning powdered metals is actually created by photons. When heated, electrons enter their excited state, and when they drop back into their normal states, they release photons, which we see as light. Different levels of energy produce different wavelengths of light, which is what causes the colors to be different. This is true for transition metals, who also exhibit color when they dissolve into their ions in aqueous solution.

Some interesting colors to note:
Barium: green
Copper: blue-green
Sodium: yellow, and often cobalt blue is used to offset this yellow color
Lithium: red
Strontium: brick red
Calcium: orange red
Zinc: colorless/white
Potassium: lilac
Rubidium: red purple