My memory is foggy at best as were my eyes most times I entered or exited, but I believe it is (or was) Nector's. Do they still serve Poutine downstairs? They had some of the best hot turkey sandwiches ever. It's a one-stop shop - food, drink, live music.
For an alternative nightspot, try the Chicken Bone Cafe - it's unique. College crowd hangs at Last Chance Saloon. I could never afford What Ale's You, but my rich friends took me there a few times. Seemed a little stuffy to me, but maybe my perspective has changed some. Last time I was up there, I had a beer at the Vermont Pub & Brewery and another at an Irish Pub on Church Street. You can MapQuest these to find them. When I went to school in the 79-83 time frame there were reputed to be 64 bars within 1 mile of campus. Never counted 'em all, but it seemed like an underestimation.
For a real Burlington diner experience, go to Bove's cafe on lower Pearl Street. Unbelieveable Italian food. Henry's Diner is (or was) the place for breakfast - it's on Bank Street, less than a block west of Church St. For higher end fare, Leunig's Bistro on Church Street was a popular spot...I could never afford that kind of food, so I had to take people's word for it.
I would avoid the chain restaurants. You can go to them in Generic-ville any time. There's too much good food and drink in B-ton to waste your time with Olive Garden, Red Lobster, etc.