Every single person I talked to recommended Tipitina's (Tip's for short). At one time it was supposedly the top performance venue in the city and it still holds it own. The original owner was a famous R&B performer and you see a picture of him over the stage. The place plays up its history as much as possible and you can see a New Orleans music hall of fame here.
So if you want to catch some really good live music, this is a great choice. When you walk in you see the bar and a large area for concerts. There is also an upper level that looks down on the stage. The night (Saturday) I was there, it was packed. I don't remember the name of the band, but they were well known to the local crowd. They played some hard rock, with a cajun twist and they were very, very talented (unfortunately this all meant a $15 cover charge).
There is also a location in the French Quarter (it is supposed to be much more touristy), but you want to go here to see the real deal. Some really big names still perform here so check the schedule. You might get lucky.
If you do go, it is about a 10-15 minute cab ride ($12) from the French Quarter.