What two colors are missing, options are black, blue, red, white, and purple. The two missing colors can be different or the same.
The board starts out as a chess board with black and white tiles diagonally. Some of these are replaced with blue or red or both which is purple. The distance between red and blue tiles is the same backwards and forwards. There is enough information to reverse the pattern and confirm whether or not those tiles should be blue/red. They both correspond on the first blank, but neither do on the second. Therefore, the second one must have been left white.
Edit to address Joe's question
Going from left to right in each row, the distance is the number of squares between each occurrence. The blues have one pattern and the reds have a separate pattern.