An atom will emit light if an electron is losing energy (the lost energy comes out as the light). It does not have to be to its ground state, but it does have to be from a higher state to a lower state. An electron moving up a state in an atom will absorb light, so you will be able to see it if you look through a filter or something else that blocks light and has a color, since only certain colors will be absorbed.

The answer is that both are possible. If you see a particular color, there are two possible reasons the light has that color.

The first case is that you began with white light from the sun (or something similar), and some colors were absorbed. You're left with light of a different color that you see. In this case, the absorption happens because an electron in a material absorbs a photon and moves from the ground state to the excited state. The color of light absorbed depends on the energy difference between the ground and excited states.

The second case is that you produce light of a particular color (say like a green laser, or a red LED). In this case, the light is produced because electrons in a material are moving from the excited state to the ground state, and emitting a photon of a particlar energy (that is, emitting light of a particular color). As before the color of light produced depends on the energy difference between the ground and excited states.