It sounds like you have Eastern Tent Caterpillars, Malacosoma americanum. This is a very common pest in the U.S.. You can cut off affected branches, and destroy them. Make sure the caterpillars are killed. You can also use a broad-spectrum insecticide early in the season. This may not penetrate the web, so it isn't the most effective method of treatment.
There are certain parasitic wasps that will attack these worms, also. You may want to research that as an alternative control method.
I recommend using more than one control method, for better results. I have also found that Eastern Tent Caterpillars rarely become a serious issue on the same plant several years in a row. They'll generally be very bad one year, and only moderate or mild the next.
The shrub/tree you are having trouble with is probably in the rose family, because those are the main host plants for these pests, although they will attack other plants.
Here are more resources on Eastern Tent Caterpillars:
About Insects, How to Control Eastern Tent Caterpillars
Bugspray, Inc., Tent Caterpillar Control
Planet Natural Research Center, Tent Caterpillar
Missouri Botanical Garden, Eastern Tent Caterpillar
Rodale's Organic Life, Tent Caterpillar
Gardening Know-How, Home Remedy Tent Caterpillar Control