Concord grapes are one of the most vigorously growing grape varieties and require routine pruning each year to control size and production. Grape pruning requires a detailed description, so refer to these two publications for direction on this job: Basic Principles of Pruning Backyard Grapevines and Growing Grapes in Kentucky.
Pruning a neglected vine: Old, neglected vines (usually in small home plantings) often are not vigorous and have several crooked trunks and considerable amounts of old wood. New canes are scattered over the vine. Much of this old wood and some of the trunks and new canes probably should be removed. Keep the healthiest canes with wood and buds matured the previous season to bear fruit. These may be selected almost anywhere on the vines. To bring the vine under better control, save enough of the best canes near the base, or at least on a central trunk, and remove all other growths, both young and old.