The two restaurants that are best known for game are the Gun Barrel (in town) and the Gamefish in the Snake River Lodge at Teton Village. Game dishes are the primary attraction at each one. I have to honestly admit that I've never eaten at the Gun Barrel, so I can't give you a personal review. I have eaten at Gamefish and liked it very much.
Our two favorite "nice" (that means higher up on the expensive scale) restaurants in Jackson Hole are Stiegler's on the Teton Village road at the JH Racquet Club Resort and the Blue Lion in town. Both have game items on the menu (the Blue Lion often has an outstanding elk dish).
Another one I like a great deal, and I love their above-the-bar wine rack, is Vertical in the Best Western in Teton Village. They often have a selection of game dishes as well. Burke's Chop Shop in town also has great steaks and quite a few game dishes.
The three most expensive restaurants in JH are the Snake River Grill and the Rusty Parrot (both in town) and the West Bank Grill in the Four Seasons at Teton Village. I'm not personally a fan of the Snake River Grill but it gets more out-of-town press than any other restaurant here. We occasionally go to the other two for a special occasion - both are really great restaurants.
If you're looking for a burger, I still believe that Billy's Burgers in the Cadillac Grill (next to the Cowboy Bar on the town square) has some of the best burgers anywhere. I love the burgers there and the place always reminds me of that old "cheesburger, cheeseburger, cheeseburger, Pepsi, Pepsi, Pepsi" skit with Ackroyd and Belushi on Saturday Night Live.
And one final thought if you're looking for a happy hour and it's a sunny afternoon. The Granary Bar at the Spring Creek Resort runs a daily drink-and-food special from 5:00 to 7:00pm. It's up on a butte way above the valley floor and the view out of their bar is probably the best view in Jackson Hole. My wife and I go there fairly often, so give that some thought as well.
I'm hungry now. I think I'll go to lunch.