What should you see in a photo to call it great?

It may be hard to give a full answer to this question as there are so many genres of photos.
What qualifies a nature photo as great may not apply in the case of facial photos, and so on.
I will try to answer for photos that capture life experiences.: A shark eating its prey; people looking at something about to happen; a baby enjoying a candy, and so on.
A great photo is one that fully captures the life in the scene.
It somehow captures the motion of the scene and displays it in the confines of the standstill format.
I admire a photo that drives me to effortlessly imagine how the picture plays itself forward into the next scene.
It captures the event right at the point when the life in it is at its climax - the facial expressions are most intense, the motion is most unpredictable, and the event is about to unfold.

Capture the subject in natural form, color, lighting, angle, moment, camera and accessories, the photographer and his creative thinking.
It's a combination of multiple things put together.
The best photographer would not have to retouch his image.

Hi Sa'ed Hudaib, Interesting Topic! I would see A B C to call it Great Photography.
A for Alignment perception B for Balancing concept C for Composition principle

Regradless of the subject of a photo, a photo or a visual should be attractive not descriptive.
Imagine you are visiting a photo exhibition.
You will be able to see probably dozens of hundreds of photos by different photographers or artists/ designers.
Out of those hundreds of photos you would only admire a few and spend more time in front of them.
They would probably be on different subjects from a desert sun set shot to a smiling baby but the overall measure to categorize a photo as great is how attractive it is to you!

I wish you had mentioned what type of answer you where interested in, there are at least two sides to a photo: a) things that help/compose or manage the visual aesthetics of the shot and b) those that help convey some meaning and has some philosophical meaning to itself.
you can have a very bad photo (aesthetically) and it becomes a great one, just because it reveals some deep meaning or make an event known to its core, and you can have the best composition and colors and contrast and absolutely nothing to show for! it would be like a golden photo frame with no photo inside! The greatness of a shot comes not from when, where or how you tried to capture it, but from WHY.
if you try to think too hard on the methods or equipments or even rules, you will clog the path between your eyes and your creativity and therefore your eyes will decide and your shots will become empty frames.

A picture that has some story in it and that takes you to the imaginary world where you become a part of that story.
A great landscape always makes me to experience something like that.
Whereas a great portrait gives me insight to the very soul of that person.
And great portraits always has great eyes in it.

I'd say a great photo tells a compelling story.
Very simple really, but not always easy to achieve.

depends, but i would say Quality

التوازن البصري visual balance

When the photographer transfers his emotion of the moment of shot to the viewers looking at the photograph successfully, that's considered as a great photo.