Brewery and Brewpub - local beer
First, a distinction. I assume the topic is for an establishment that brews it's own beer and serves food prepared in the same establishment.
In that case:
Pittsboro, Carolina Brewery - for great local beer and good food - Flagship IPA and seasonal Oktoberfest are my picks.
Chapel Hill, Top of the Hill - good local brew and decent menu - great location on Franklin St.
Can't really tell you about Raleigh, Apex, or Durham.