S.G.
Since the area that is brown is dead you will need to get a rake and rough up that area, break up the soil a little. Then get some grass seed, ask your local home improvement store what kind is common for your area. Then sprinkle that over the area and water in the morning and the evening. In about 1-2 weeks you should see some grass coming up, just be careful not to mow it until it is at least 3-4 inches tall and don't walk or play on the area until it is established. Fertilizer won't do any good since there isn't any grass to fertilize in that area.
Good luck it isn't as big a deal as you would think. I don't have a green thumb and even I can grow grass. Ha Ha.
Michelle