Well, mostly, they wave their arms and get paid a lot. Like most senior managers (good and bad), they don’t seem to do much at all.
In this TED talk, Itay Talgam uses some wonderful examples of conductors at work to show how less can often be more, and even how nothing at all can be everything. I found his last example quite awe-inspiring.
He talks about Strauss’s Ten Commandments of Conducting, which can be found here (or here).
Thanks to JD for sending me the video. I’m sorry it took me so long to watch it.
Proper Discord 
I’ve always thought most of what the conductor does is done before the concert itself. Finding elusive, hidden voices, creating new interpretations and helping the orchestra bring it to life, inspiring, rejuvenating, communicating with the orchestra and consequently (hopefully) the audience. All of this should be done in rehearsal so that on the night itself he (or she) is really just gently directing a group of musicians who already know exactly what needs to be done thanks to the hours they’ve already put in with the conductor. They seem to do not a great deal, but I think behind the scenes the great ones (Dudamel, Minkowski, Jacobs, et al) are doing an enormous amount of preparation. Or am I being naive?
And remember: never look at the trombone. It only encourages me.