The basics are surprise/rarity, emotional impact/narrative (which can include comedy, tragedy, drama, etc), geometry and composition (which can include focus control, depth of field, and motion effects), and controlled colour.
Colour control is the one a lot of people miss. The aim is to have a limited colour palette - either one dominant colour, or two or three punchy related colours (complementary or contrasting on the colour wheel) against a more neutral background (white/black/grey/less often - brown.)
Too many colours are chaotic. There are exceptions which can make chaotic colour work as a feature, but if you just snap a random crowd you'll usually get a photo that lacks something, and you're not sure what.
Newspaper photo pages are good for examples. Random sample:
You can see how few colours most of these photos have and how often there's a strong colour contrast. A "newsy" photo of flooded buildings still has controlled colour - base of green contrasting with blue in the foreground, and further bright purple/pink contrasts in the background.
And the photo of the soldiers with masks balances black/white against the pale blue of the masks, with complementary gold highlights.
Even the prisoner photo only has three colours - green on the prison shirt, pale yellow gold on the uniform patches and buildings, and some splashes of red, which contrasts with the green.
The opera audience photo is the weakest with a pale nondescript yellow/green wash against grey/white, but it (almost) makes up for it with very strong geometric composition in the architecture.
Edit: non-representational photos usually follow the same rules. The one big exception is high end art photography which uses a couple of specialised standard palettes (usually warm on the flesh tones) and a small selection of standardised subjects that signify "art" in that market.
And too suggest that art photography uses specialised standard colour palettes to signal “high end art” is a pretty big claim to make with no evidence.
Some of the most influential and interesting art photographers working today produce images that do not exhibit the qualities you mention, for example Jeff Wall and Thomas Struth.
IMHO, what makes a picture good is a combination of aesthetics and narrative. The idea that the quality of an image is subjective reminds me of Robert Pirsig's pursuit of the notion of "quality" in his two books . Although it is very hard to define the rules that govern what constitutes quality, if you survey enough people they rank simple artifacts like images quite consistently; perhaps differing when the narrative makes a significant contribution.
This post presents some very high quality images. What is missing is the photographic context that contributes to the aesthetics of each image.
> What makes a picture evoke emotions in – or feel “real” to – us, is what we see in it – based on our education, knowledge and history.
> There may be pictures from your past and when you look at them you are reminded of those experiences and therefore they feel special to you, like this beach picture above: it was one of the first analogue pictures I took and I somewhat like it, but it is really that good? Probably not.
Same goes for pictures of your girlfriend, children or dog: they will all be special to you, but to someone else they may not be that interesting, especially if they are lacking in one way or the other (out of focus, tilted, noisy etc.)
> If on the other hand you want your pictures to appeal to a broader audience, you should make sure they cover an interesting topic that means something not only to you. And they should also be good in a technical way, which brings us to the following section.
Exactly, f/8 and be there. 
Likewise, photography is a broad discipline ranging from pictorialism to postmodernism and a good photograph is more than opportunity, emotion and execution.
Also, I wonder how much of professional amazing looking photos are edited (e.g., brightness, contrasts, levels, saturation, etc.)
As for editing for brightness, contrast, ...: All digital images are edited. You either let the camera manage levels & color ("straight out of camera") or you do it.