When casting, I usually bring in actors two at a time and have them read opposite each other. I try to match people up in ways I think might work -- an admittedly hit-or-miss process when you're just looking at the headshots and resumes. When the actors actually show up, sometimes they seem mismatched, but I can still get a good impression of each individual actor, even if I know I'd never cast them together.
By the time callbacks come around, I usually have definite couples I want to see together. Sometimes I'll be pretty sure that I want to cast, say, a woman in a given role. Then I might have her come in for an hour and have three different guys work opposite her, one after the other. Or I might have a group of actors come at the same time but see them only two at time, in different groupings. These are just different strategies to be able to see people I like working with as many other people I like as possible.
In the end, I need to have seen the people I want to cast work together -- I don't cast actors separately -- I need to see how they work together, act around and react to each other. Chemistry, ya know?
A few general casting tips:
I usually do a brief warm up with the actors -- a stretch, a voice thing, something silly and relaxing to loosen them up. Most directors don't do this, but I want the actors to be relaxed; I want to get their best work.
I actually try to direct a little during the audition. When I give the actors their sides, I'll tell them a little about the characters and what they're trying to achieve during the scene. Then, after they do the scene once, I'll usually give them a few notes and have them do it again. Actors love this -- they want to do better. And it's good for me to see how well they react to direction.
I usually never stick solely to the screenplay. I'll have the actors read a scene or two, but I'll also give them an improvised scene to work. Why? Because I might want to see the actors deal with certain emotions or situations, and the actual scene from the screenplay may not be written in a way that makes it useful in an audition.
Also, some actors show things in improvisation they don't show in cold readings -- and I want to see what they're capable of. The setups for these improvisations have to be thought through, though. You have to know what you want to see and set up the improvisation accordingly.
For example, for "Mouse" I needed to see if the actors had the capacity to relate to each other well as lovers. The screenplay deals with their characters' failure as a couple -- there's not much in there that shows them working together. But in order for that failure to have any importance, I knew that the audience had to feel the intimacy that's being lost. Now there was no scene in the screenplay that would let this intimacy come out very well... So I had the actors improvise their characters' first date. It helped me see what kind of depth the actors had, to see how they were conceiving and breathing life into the characters.
I try to have the auditions taped -- by someone competent with a video camera who's able to follow the action and zoom in for closeups from time to time. A static wide shot of the whole room won't tell me what I need to know -- I need to see the actors' faces -- their eyes, really -- to know if they'll work on screen.