Remove the old zipper from the jacket. You will want to remove both sides of the zipper, even if only one side is defective. Unzip it and turn to the back side of the zipper. Using a seam ripper or a small knife, begin cutting the stitches without slicing the jacket. Remove both sides of the zipper in this manner.
Purchase a new zipper from a fabric or shoe repair store. Take the old one with you so you can be certain to get the proper thickness and length to repair your jacket.
Zip the new zipper and lay each piece along the front of the jacket where the defective zipper was. Once you think you have it lined up straight, take some safety pins and pin each side individually to the jacket at the top, middle and bottom. This will help make sure you sew the new zipper on straight, so it will zip properly each time you need to use it.
Unzip the zipper and begin sewing each side individually to the jacket as the original was. Remove the pins once you have that side securely in place. Repeat the sewing with the opposite side. Using a sewing machine will ensure a straight, even stitch. However, if one is not available, a needle and thread will suffice.
Try on the coat and test the zipper.
