What does it take to be a good accompanist? I'll start with with an accompanist is not: a timid soloist. Unfortunately, sometimes in school it looked like pianists who weren't assured enough to be soloists (meaning they got too nervous about being listened to by themselves) were made accompanists, the way violinists who couldn't quite make it sometimes were given violas. Neither situation works very well.
In fact, I recall much complaining about how some of the pianists in the Van Cliburn International Piano Competition weren't very good at the chamber music round because they weren't much for playing in an ensemble. In this case, we are looking at the reverse of the idea in the paragraph above: that it is soloists who are in fact inferior accompanists!
Neither of which ought really to be true. What makes a person a good accompanist is really the very same thing that makes a person a good soloist, or just a good musician, period. In a word, it is this: listening.
A soloist listens to him or herself--for balance between the voices, for the acoustics of the hall, and the clarity of the pedaling, and is constantly adjusting to that external feedback. An accompanist is doing the same thing, only now the task includes notes that they are not playing as well as notes they are.
In the case of an accompanist, particularly if you are working with a conductor, this means surrendering your own inner metronome, while at the same time having a good sense of rhythm so that your playing is clear and rhythmically precise--but the moment the conductor makes an adjustment to the tempo, you have to make that adjustment yourself. In other words, the 2nd beat of the measure only comes when the conductor decides it does, not when you think it ought to. This is probably the thing that causes most people to be poor accompanists, because it means constant attention to a force outside of yourself. I'll give two examples of what I mean, both from the Cleveland orchestra.
When I was in college, back before the turn of the century (I'm now an ancient 42), I used to get to hear the Cleveland Orchestra for free every weekend because at the Cleveland Institute of Music if you were one of the first 40 students to sign up you got free tickets. I learned a lot watching one of the great orchestras of the world and its excellent music director, Christoph von Dohnanyl.
One night an insufficiently prepared soloist did something very rare on the stage of that august hall. He skipped two beats in the middle of a solo section. The conductor reacted instantly. Having been trained in opera, he gave a very fast four beat pattern, smaller than the usual beat pattern, in the manner that a conductor might do if a soloist were singing a recitative portion of an opera to tell the orchestra that the downbeat of the next measure could come at any time without counting out the regular four beats in sequence. Without any ceremony he therefore rushed through the next measure's beat pattern in time to give the downbeat to the following measure right in time with the soloist. This meant that within one measure of the pianist's skipping those two beats, the orchestra was right back with the soloist. The entire orchestra, watching their conductor like 70 well-trained hawks, nailed the sudden change, and I doubt whether anyone in the hall who didn't just happen to be playing the very same concerto with another orchestra later that month (such as myself) and thus know the piece really really well, would have heard anything wrong whatsoever. Amazing!
The next example is even rarer: when the maestro himself made the only mistake I ever remember seeing him make in a rarely performed 20th century American symphony, the entire orchestra had a momentary (but oh so short) collapse--he failed to give a clear second beat in a single measure and the whole orchestra was depending on it. Now if they hadn't been that tuned in to their music director, and had all been operating on their internal metronomes and not paying him that much attention, that wouldn't have happened. Ironic that it was their greatness that caused their downfall! But, like I said, this miasma lasted for only a fraction of a second. So many other golden moments occurred because their own sense of rhythmic precision, as finely tuned as it was in the case of each individual player, was completely given over to the maestro. Where he put that second beat--slightly later or slightly earlier as he sculpted each phrase--that was were the entire orchestra put it, to a player.
I have plenty of my own stories about skipping beats are adding beats to accommodate soloists or choirs or conductors, most or which are not glamorous--but they are good training! The most training I ever got in a single evening involves a young woman who was not prepared for a school recital and kept skipping ahead or behind as her memory failed her: now a line ahead, now a page back--causing her nervous accompanist to have to find where she was in the music as quickly as possible. Now that's accompanist boot camp! (She failed the recital, of course.)
There are many situation that require being in sync with an external force. You can't be a good outfielder unless you put your glove is where the ball is, not where you'd like it to be. Until then you have to constantly track where it is headed. It is not dissimilar with the conductor's downbeat. You can find it if you are tracking the preparatory beats and making adjustments are needed. That requires a good deal of refined focus. It is being able to meet lots of little deadlines--only, they are not marked on the calendar, and they are continually subject to change.