The quality of an image can be judged in three broad categories:
Here we have factors like sharpness, exposure, color, white balance, depth of field, noise/grain, etc.
Compelling emotion or insight
The photo must elicit some emotion that makes it memorable. A compelling photo provokes an 'aha' experience, hard to describe but easily recognised.
This is what we usually call composition. It includes things like line, shape, color, tone, balance, perspective, etc.
A good photo will score well in all three categories. If you must rate photos then I suggest you score them on a scale of 1 to 5 in each category. The big question though, is, does each category have equal weight? My personal preference is to give more weight to 'compelling' photos. My advice is not to get too detailed about individual aspects of the three categories. go by your overall impressions of Quality, Compelling and Aesthetics.
Thomas Carlyle said 'There is nothing more terrible than activity without insight'. He could easily have been talking about photographers. The bottom line, for me at least, is that a good photograph provides a compelling insight.
Just to round things off, here is a counter-intuitive view of the subject