You won't find any Broadway type shows here, but it's a great theatre nontheless.
Yale Repertory Theatre has a wide array of theatre that is performed. They average from student shows to professional shows by famous author/writers.
The theatre is a decent size and the tickets are relatively cheap for a theatre in New Haven. Parking can be a problem (as any place in the downtown area) so plan on getting there early to scour the streets for on-site parking.
Shows typically start at 8:00pm, but check the listings as some venues offer different show times.