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Prepare the damaged area by removing all dead grass and 1" to 2" of soil 3/4" past the edge of the dead grass.
Water the area, then amend the soil by working in either compost, leaf mold or fine-milled peat moss to a depth of 3" to 5". Be sure to add enough to make up for the soil you removed (if the area is large, you may want to roll it with a half -filled water roller).
Sprinkle the seed over the spot according to the rate on the box. Many seed boxes have examples of what the right density of seed will look like; if yours doesn't, be careful not to seed too densely. It's better for seed to be a little too thin than too thick. Also, although it isn't always necessary to match the variety currently growing in your lawn, try to select a seed variety that is similar in color or texture to the existing lawn.
Cover the seed with a thin (1/4" to 1/3") layer of your amendment and tamp the soil. If the patch is large, you may want to roll it again with a water-filled roller.
Water the area with a fine-mist nozzle. Keep the area moist until the grass emerges, then gradually back off your watering over the next week or two.
Note: You may need to lengthen the watering if you live in a hot, desert climate.