My little boys are watching Kim Possible right now and in the credits a director is listed, along with the producer and animators, etc. What does a cartoon director direct?
They are like the project manager for assembling it, I believe.
The same thing a live-action director directs: the actors.
The director in any TV or movie is directly responsible for making sure what we see on screen is what needs to be seen in the way they want it to be seen.
They direct the animation supervisors nad the voice actors to make sure his or her vision is fulfilled. They also sign off on designs and “locations” (backgrounds) and all the other things live-action directors do.
It can be quite a challenge to direct what is essentially a frame-by-frame element-by-element process, so for feature films they will often have co-directors to assist.
An animation director has to decide what will be in each scene, how long each scene will last (and in cartoons these decisions work down to fractions of a second) and where the cuts will be between scenes.
To use the Kim Possible example in the OP, suppose the script says, “Ron and Rufus both slap their foreheads in reaction to Kim’s line.” The director would have to decide if he wants the line and the reaction to all be a single frame with all three characters or if he wants to cut from Kim to Ron and Rufus between the line and the reaction shot. Does he want an upper body shot or a close-up of their faces? How many frames should the reaction shot hold for? Should Ron and Rufus slap their heads at the same moment or one after the other? How much should they move? What noise should they make? What should their facial expression look like? What should their postures be? What should Kim’s facial expression and posture be if she’s in the shot? These are all decisions that have to be made for a single line of the script that’s going to be about a second of screen time. And unlike live action, the director pretty much has to decide everything in his or her head before the shot is made - they can’t simply try it several ways and see which one works best on screen.
My knowledge is more of the old Warner Brothers cartoons than current TV cartoons, but what I know of that is in accordance with what the others have said. The director is the guy in charge. He (still almost always a “he”, I believe) directs the voice artists, deals with the music guys to make sure that you hear what he thinks is funny, cracks the whip with the writers and re-writes as necessary, tells the animators what he wants and makes sure the timing works. Since he will normally have been a crack animator himself, he may do the “key” drawings for the cartoon, and have the other animators work around them (Chuck Jones was well known for this at Warner Brothers). Despite the jokes, he does not say “OK Elmer, take it from 'You wascally wabbit”.
Watch any Pixar movie’s making-of content. Even a live-action movie director does much more than just sit behind the cameraman and yell “Action!”
The director is in charge of the film from story, writing, and storyboarding* through voice recording**, backgrounds/sets, animation (including final approval of each sequence) and final assembly of the pieces.
*: Animated films are “edited” at this stage. There is very little decision-making at the end as is the case in a live movie.
**: Western animation records dialog first and animates to that, Japanese animation usually is done first and dialog recorded to it.
Andrea Romano is credited as “Voice Director” for many of the DCAU 'toons, as well as on some Pinky and the Brain episodes.
And while researching this, I see there’s a direct-to-DVD Justice League movie coming out soon and Lucy Lawless will (finally) get to play Wonder Woman.
Although if WB ever does another Roger Rabbit/SpaceJam type mixed cartoon/live action feature, wouldn’t that make a wonderful scene? 
At Warners, and I think at some ther studios, they were not even known as directors until well into the 1940s. Warners cartoons before then credit “Supervision.”
In the Japanese animation world, the director will often draw the storyboard, but will at minimum oversee and approve it’s creation. If they are the lead-director, they’ll often have the same power over the screenplay. As the animation is being produced, they are shown completed keyframes and will make corrections that the person has to go back and fix.
It really sort of varies by director and the size of the project. Some will direct the voice actors, others won’t. Some will select and/or approve the music, others won’t.
Thanks, everyone! I understand pretty well what a live action movie director does, but I couldn’t quite wrap my mind around someone “directing” animated characters.
And I must tell my husband that Lucy Lawless is going to light up the screen again. He and the gassoons used to love Xena!