Lots of people will be able to recognize a great photograph when they see one, but they won’t always know why it’s a great photograph until someone like me points it out in the most boring way possible. You’re welcome!
Anyway, one of the most important characteristics of a great photograph is composition. The balance of elements of the photograph is important. Photos with the subject smack dab in the middle of the picture are ugly to look at, even if the subject itself has any visual appeal. The subject may be nice to look at in this case, but the picture itself isn’t going to be.
A better composition involves having the subject in the right or the left side of the photograph, with enough negative space surrounding it that it creates a certain degree of balance. The subject is still going to dominate the photograph as long as the photographer emphasizes its size or its lighting in the right way. However, this sort of composition overall makes for a vastly better photograph.
Getting a good contrast of light and shadow makes all the difference. The photographs that are visually compelling are the ones that really draw your focus in when it comes to the darker areas of the picture. Photographs that have flat lighting where everything looks exactly the same and there seems to be no shadow in the implied world of the photograph are not as interesting to look at as the photographs that are striking.
The photograph can’t be too empty and it can’t be too busy, either. Too much business is going to distract from the subject that you are trying to capture. If the subject is the only thing in the picture functionally, then the picture is going to look to flat and even too fake.
Establishing the balance of all of these elements is going to make photography sound hard and it is. However, it will eventually become intuitive for all people to balance all of these elements, and they will soon be able to recognize good and bad photographs instantly, for better and for worse.