Q.What should I do about a dead tree branch?
My peach tree is several years old and has 3 branches coming out of the trunk. This year one of the branches did not get leaves.

Here is a link that demonstrates how to properly cut off a tree limb using a three step technique:

Here is a couple of links that will help you determine the best time to prune and trim away the dead branch.
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-care.htm

Cut the branch off as close as you can to the trunk as possible and maybe paint the cut with some undercoat paint to prevent the tree losing more moisture. Cut up from underneath first,about a third of the thickness of the branch, then finish by cutting down from the top. This will prevent bark from peeling off as the branch falls to the ground. Hope this is of some help.