I'm gonna be "That guy" who talks about how the utilization of rhetorical devices, and command of the English language. I think that those two are the most important foundation to a good story. If at your base, you don't have much strength to propel the story, it's doomed to fail. And I mean this before characters are even involved. You can imagine the most heart-wrenching scene in your mind, but if your manipulation of diction isn't there, the scene won't be nearly as powerful.
I recently read Metamorphosis by Franz Kafka, and I absolutely loved it. Yet, literally nothing happens in the story. All of it is internal, and the story takes place within a single room. I realized, around halfway through, that I actually loved the story. Kafka's manipulation of his words made it such a good read, and it kept it refreshing. I don't want to spoil it for anyone interested in reading it, so I will leave it at that. I just think that this is a good example of how important the implementation of rhetorical devices and the use of the English language can be.
Then, when the characters are added, it's important to have fleshed out characters. A lot of people remember stories for their scenes. For instance, I like to think back on the scene in Halo: Reach when Jorge threw Noble Six out of the space vessel while he stayed back and detonated it. This scene fleshed out Jorge's character in its final moments, while propelling the story forward. Over time, I've realized that characters have to keep the story moving, as they aren't standing with a backdrop changing behind them every few minutes.
Emotion is an important thing, as well. I don't just mean sad, though. I mean that there needs to be the highs and lows, the characters need to reach the breaking point, and either admit defeat, or emerge stronger. You can't just have characters meander through unscathed; they have to evolve as the story progresses. It's important to be able to look back and see how they've changed, whether it's for better or worse. I think Herman Hesse's Siddhartha did this well. He began as an arrogant kid, searching for enlightenment, but ended as an old man, finally wise enough to find nirvana.
Above all, there needs to be a conflict. If there is no conflict, then no character development will ever happen. It will just be masterful use of the English language that only the hipsters in Starbucks will appreciate. Conflict is the catalyst that begins the evolution of characters, and keeps the story moving along. Without it, the story isn't going to be much.
For a long time, Shakespeare's Julius Caesar seemed like a story that was just conversations that led up to the death of the protagonist, who is also the title character. It wasn't until I realized that Marcus Brutus is the tragic hero that I saw the true conflict. It wasn't the story of killing Julius Caesar, it was the story of a man at war with himself because he must choose between killing his best friend, or seeing him turn into a tyrant.