What does the colors in tartans mean?

The tartans have evolved considerably from their beginnings.

Wool from the local sheep was spun and dyed using the natural dye plants of the area the spinner/weaver lived in. No one in those days could afford to import dyes.

Plaids/tartans used a number of various colours in their make up.

Partially because dyes were primitive enough that each vat of dye was a little different from all the others and partially because many natural colours seen at a distance, blended to make sort of a Celtic camouflage. Remember these clansmen were hunters; if for no other reason than they needed to protect their livestock from predators.

As each household made most of their own cloth, the patterns were handed down within the family. This is most likely why clan tartans evolved.

Because of the distances between clans and even families there was no specific "meaning" for each of the colours, they tended to use what they could get, not search out specific dye plants.

It is only in modern day that clan tartans have become formalized. If you check you will find there are still "hunting plaids" listed.

There are however, to my knowledge, no formalized meanings for the various colours within a tartan.