There are currently three dinner shows available.
Hoop De Doo Revue (Fort Wilderness): Guests are seated on the floor of the hall, or in the balcony. All of the chairs swivel so you can turn to the stage to watch the show. It's a 6-person show done in the style of a western traveling vaudeville show, complete with corny jokes. Food is served at the tables, generally between "bits" on stage, and is part of the show itself. Includes wine and beer.
Mickey's Backyard Barbecue (Fort Wilderness): Outdoor meal at long picnic tables (not reserved seating), with "picnic" style food, wine, and beer. Instead of a "show", there's live music, some acts (like rope tricks), and Mickey and friends will come out onto the "dance floor" area and mix with guests. This is not considered a normal "character meal" as the characters will not stop for autographs but rather just join the dancing.
Spirit of Aloha (Polynesian): I know less about this one, but it is, I believe, amphitheater-style seating with a Hawaiian luau theme for food and entertainment.