What qualities do great conductors share? Are there unique qualities that make a particular conductor great that do not exist in other conductors?
The conductor's job is to analyze the piece and figure out, to a very minute detail, how it should be played. He then has to communicate that in rehearsal through words and physicality (conducting patterns and gestures). The difficulty is in coming up with a good analysis, having a sharp ear to hear what the ensemble is doing, and being able to communicate effectively. A truly great conductor can imagine himself in the shoes of any individual musician in order to innately understand what each person needs from him.
Orchestral conductors also should ideally know enough about how every instrument is played to more intimately understand the difficulties of the music, and the tendencies that are going to arise as the music is played. Most of this work is with the strings, as there's more variation in approaching the music (part of the bow used, playing closer to the bridge or fingerboard, specific bowing techniques, and even which strings to play things on) than with winds where there's usually only one reasonable way to play anything. For example, if you notice that the violins' attacks are muddy and you see that they're playing in the middle of the bow, you could ask them to move closer to the frog.
All of that being said, the biggest mistake I see with newer conductors is a simple lack of preparation and command of the material. I would so much rather play for a conductor who had clearly studied the music and knew what he wanted, but was a bit clumsy with his communication (both verbal and physical), than for one who was masterful with the baton but clearly hadn't done his homework. And I see a lot of beginning conductors place too much focus on the stick-waving and underprepare the actual analysis of the music.
The key qualification is a very analytical ear and a well-established idea of the score (with lots of experience under one's belt, one will be able to work with less preparation, but it takes a long time before that).
To work with a large number of individuals with different problems, it is very important to hear what stuff does not work properly, find effective ways off working it out, and put it together again afterwards.
I was part of the training material for a master course in conducting once. It was absolutely flabbergasting what kind of difference a good conductor made to the quality of the (rather good) choir. With a conductor who did not really manage to do more than repeated runs through the score, the level of the choir was several classes below the potential a good conductor (also without previously knowing the choir) was able to work out from quite harder material, just by knowing exactly where stuff went wrong with what group and/or combination and how to work it out.
Mind you: all the time it felt for the singers that they were being dull and/or able to pull stuff off. You really did not realize the impact of the conductor without the actual side-by-side comparison.
You just had a bad day with the first one, and a really good one with the second. Except that it was one right after the other.
Good conductors are good leaders. They know the score and how it's supposed to sound. They need to be able give constructive criticism to each and every musician. They also need to be confident and good analyzers of both the score and the players and the sound being produced.