To highlight just a few seems unfair. But you probably have only 4-5 days to visit, so it makes sense to give you a small group of theaters to get you started.
Just west of the Loop on Division Street is Chopin Theatre, home of the recent smash hit production of Our Town, which wound up on Broadway. The basement theater is always buzzing with something new and different.
To the north, a new theater district of sorts is turning up. There's Theater Wit and Stage 773 (the neighborhood area code), where several local theater companies (over 13 at last count) are performing, such as Porchlight Music Theatre, which is committed to producing musicals with a decidedly Chicago flavor.
Nearby is the TimeLine Theatre Company, which features plays inspired by history, such as their award-winning Frost/Nixon, A Walk In the Woods, Firoello and The Front Page.
A little farther north, the Steep Theatre Company crams more tense drama into a fresh, newly redesigned space than you can imagine. It's drama up close and personal.
To the south in Hyde Park, at the University of Chicago, the well-established Court Theatre combines powerful and sensitive acting with sharp, meaningful material like Gershwin's Porgy and Bess and Invisible Man.
That should get you started on the right theatrical foot.
One more stop when you get to town might help: stop by a Hot-Tix location, for day-of discount theater ticket service. Their daily list of possibilities will be tempting regardless of your taste in theater.