In Plain English, What Are the Within and Between Class Variances in Otsu's Thresholding?
i'm avoiding labels light and dark because you could say that we choose 128 as threshold which perfectly separate light and dark pixels but consider you have a picture with only two gray levels which are 240 and 250 and you want to label them but both seems to be light.

now consider an example which you have a part of x-ray image which composed of soft tissue and bone and you want to separate them. we know bones generally have higher intensity than soft tissue in x-ray image. but some part of bones are very thin and some part of soft tissue is very thick so some pixels which  belong to bone have lower intensity than pixels belong to a soft tissue. in this problem any threshold that we choose lead to some miss labeled pixels.

lets consider more details. bones have thick and thin areas which lead to a variance around average intensity of bones, this variance is called within class variance and the average value represent the bone intensity value. the same is true for soft tissue. the difference between average of soft tissue and average of bone is called between class variance.

in ideal situation we have minimum within class variance and the averages must be as far as possible from each other (highest between class variance) to have the minimum mixing between these two groups.