Kingsgate is the main Union bar. It's at the back of Level B past Reception - it has a good selection of lagers and beers on draught and a wide range of bottles and spirits.
Kingsgate's location in Dunelm makes it a great place to meet up after lectures or possibly the best place in Durham to meet people when going out in the evening. You don't get a much more central spot for the price.
There's outside seating on the balcony and in the summer when the weather's nice there's a barbecue.