In addition to appropriateness for a role, on a basic level, directors look for three things in an audition: vulnerability, variety, and humor.
If you think of vulnerability as being "emotionally available," you'll probably understand why it's so critical to truthful acting and successful auditioning. It means bringing yourself fully to the acting experience onstage, in rehearsal, and in the audition room. To make yourself vulnerable as an actor requires preparing and rehearsing so that you connect personally with the material.
Variety is an indication of an actor’s creativity and flexibility. Unless you’re reading a very short scene, you should be able to find variations within it. Locating the changes a character goes through will almost certainly lead you to variety. Although there may not be a full arc, especially in a short scene, there should be a sense of the character moving from one point to another. If you look for the possibility of variety, you are likely to find the specificity directors are looking for.
Humor is a third critical component of a successful audition. If you know the play is a comedy, and you don’t think it’s funny, ask a colleague to help you find the humor. Comedy works best when acted honestly, when the character has no idea why what she's doing is funny. What if you’re auditioning with a scene that doesn’t contain much humor? Find a moment that might lend itself to a comic choice. If that isn’t possible—and sometimes it isn’t—try sharing a laugh with the 'room,' perhaps at a callback. Even if I’m directing a serious play without a lot of humor, I want to know that rehearsals will be playful and fun.