[researcher]

 I don't understand why copper(I) and copper(II) have different flame colors. I thought it was a characteristic of the metal, can anyone explain?

[cth]

Absorption/emission of light depends on the electronic configuration of an element.

When an atom absorbs a photon, one of the electrons goes from the ground state to an excited state. Then, it goes back to the ground state by emitting the photon it had absorbed, with the same wavelength (at least for simple cases). That wavelength depends on the energy gap between the ground state and the excited state.

The number of electrons for Cu^I and Cu^II are different. So, the possibilities to excite an electron are different  different colors.