When these terms are used to describe differences in sound, it's a direct relation to the presence or absence of particular frequencies in the harmonic spectrum.

To put this in simple terms ... if something sounds dark it's because there is an absence of higher frequencies and/or an attenuation of the lower frequencies in the timbre of the instrument. Likewise, if something sounds bright it's because there is an absence of the lower frequencies and/or an attenuation of the higher frequencies in the timbre of the instrument.

Using the words dark, mellow and bright is just one example of how someone might try to verbally describe what they are hearing. Like describing colors, sound is perceived differently ... and at times, can be very subjective.

Hope this helps!