I'm going for local theaters that are a step up from community theater? For example, I've heard of the North Shore in Massachusetts, Paper Mill in NJ, in Los Angeles, Ahmanson, Mark Taper, etc...
When you see regional credits listed in bios, they are usually in reference to any LORT theatres the work has been in (League of Resident Theatres.) Here is a link to the League's site:
Regional theatrs can be Equity or non-equity. They can be members of LORT on not. They can be Shakespeare Festivals, Dinner Theatres, Summer Stock, Outdoor Musical Theatres, you name it.
Paper Mill, North Shore Music Theatre, Goodspeed, The Goodman, The Guthrie, et cetera, et cetera.
Basically, antyhing professional that isn't Broadway, Off-Broadway, Off-Off Broadway or a National Tour is regional.