Summarize:

Hornet nests are small and easier to remove in the early spring. A nest that's around the size of a ping pong ball probably only contains a queen and unhatched eggs, so you can just spray it down with a hose. You'll need insecticide to get rid of a larger nest. By summer, nests can grow to the size of a basketball and contain thousands of hornets. At a minimum, wear a thick, long-sleeved shirt, pants, thick leather or rubber gloves, and boots. Coveralls can add extra protection, and a beekeeper's veil can protect your head and face.  If you can't access a veil, you should wear protective goggles and a hat that covers your ears. To prevent hornets from getting into your clothing, use rubber bands or duct tape to secure your sleeves to your gloves and your pants to your boots. If you've seen hornets buzzing but aren't sure where the nest is, observe which direction they fly. When they find food, hornets collect what they can and bring it back to the nest. Try leaving out bait, such as pieces of fruit or meat, then watch it from inside your home. When hornets show interest, look for the direction they fly away, then follow them. Eventually, you should be able to track down the nest. A hornet's nest is gray or beige, shaped like a rounded teardrop, and can be as large as a basketball. They usually hang from trees, but you might find one on the ground. Once you find the nest, stay a safe distance away, and mark your spot so you can find the nest later.  While you're at the nest, see if you can find the opening. Hornet nests usually have a single small opening toward the bottom. You might want to use binoculars to look for the hole so you don't have to get too close. You'll need to apply insecticide at night, so mark your spot with a brightly colored flag that you'll be able to see in the dark.
Scout for small nests on your property in the early spring. Wear protective clothing when you search for and treat the nest. Lure and track the hornets if you don't know the nest's location. Place a marker 15 to 20 ft (4.6 to 6.1 m) away from the nest.