How does electron structure of the atom and emission spectra explain the color of various fireworks?

Oh dear, this explanation needs some quantum mechanics. Oh well, here goes.

The color of a firework depends on which atoms or ions are used to give off the light. For example, Potassium ions glow red when burning, Sodium ions burn yellow, etc. Why?

Simple. every atom has a number of electrons, which occupy things called energy levels. each energy level has a different amout of energy associated to it ,and when an atom or ion is at rest (when it isn't burning, or being excited in any other way) all of its electrons will occupy the lowest energy levels they can. When an electron gains energy, it will move to a higher energy level, and the more energy it gains, the higher a level it moves to. The elctron will eventually move back to a lower level, to try to be at rest. When it does so, it loses energy, which is released as a particular wavelength of light.

So hten why does each Atom or Ion produce different colors when it burns?

The thing about energy levels in each atom is that they are discrete. Menaing htat they are not continuous, so an electron can only exist within energy levels, never between. Each atom adn each ion has differnet energy levels, so whenever electrons move between them , differnet colors of light are produced.