"Family Style" is a type of service where they set food on the table like the typical family used to sit down to in the evenings. Not the case much anymore. For example, there might be one platter of fish, a bowl of beans, a bowl of potatoes, etc, rather than portioning each plate a limited amount. Seconds are typical in "family style," but they bring more of only that which needs replenishing. The reason it isn't done much in the restaurant biz is due to the enormous waste of food/cost. As I mentioned, there aren't many restaurants with family style service in our area. Bait House at WaterColor used to try it in the evening before they closed, and Stinky's does/did it maybe once a week.
Here is a link to Stinky's "Family Style" Dinner, which they are featuring tonight. It sounds a little different than the traditional "family style" service, and sounds more like a limited-choice menu, but I haven't tried it yet, so I am uncertain of how it is served.