Getting rid of bee, wasp, and hornet nests is within the reach of a careful and smart homeowner. These insects can be dangerous, but do not have to be. First and foremost you have to worry about personal safety. People allergic to venom should not attempt to handle insects. When approaching a nest, always have an escape route. For yourself, and anyone else present. Don't approach the nest from a vulnerable position, like a ladder or the roof. Make sure you can run. Always prepare for the worst case scenario.
If you have never dealt with bees or wasps before, it's best to familiarize yourself with them. Most of the time they are not at all interested in you. A wasps circling your outdoor dinner table is interested in the food, not you. Instead of trying to get the wasp to leave, let it land on whatever it wants to land on. Inevitably that will be the dinner plate, not you.
The wasp will try to pick something up and then leave. Cut a tiny piece of whatever the wasp was after, and place it somewhere away from the dinner table and closer to the nest. If the wasps find it, they will not bother you again. They just found a food source that does not have large human shapes around it.
Don't leave the food out after you are done eating. You might attract other insects.
1) Identify the species
Begin by identifying the type of insect you are dealing with. This will help you decide what to do. Killer bees are the obvious thing to check for. If you have them in your area, you probably already got attacked. Call an exterminator immediately. If you have a honey bee swarm, you are in luck. Bees are protected by federal law. You cannot kill them. Sell them instead. Bee swarms are valuable these days. Find the local bee catcher. Most of the time the company will trap the bees free of charge. Sometimes you will be able to sell them for a small fee.
If you have a smaller nest with larger insects, you are probably dealing with wasps or hornets. These types of nests are typically removed and the insect colony is destroyed. Depending on the size and location of the nest, you may be able to do the work yourself.
2) Option: Do nothing
First, consider doing nothing. Wasps and hornets will not automatically attack you when they are out and about. If the nest is in a location where it is not disturbed by your daily routine, and not to many insects end up in your home, let the colony be. Come winter the colony will naturally die off. You can remove the nest in early spring.
I strongly recommend this route. There is a colony on your property for a reason. It is eating something. Chances are new insects will move in each year. Since insects are territorial, the ones on your property will keep others away. Learn how to live with them.
3) Option: Knock the nest down
If you want to get rid of them you need to create a plan for how to do this. Small accessible nests can simply be knocked down with a shovel. Make a quick exit afterwards. There is no need to kill the actual wasps. The nest is completely vulnerable on the group. Other critters will find it and dispatch the insects for you. If not, the wasps will likely abandon the nest. Just check a few days later.
4) Option: Insecticide
The next step up are poison sprays. Every home center carries them. Some can be sprayed on the nest from 15ft away. Apply them according to the directions on the packaging. Remember to NOT to check your work. The insects will be on alert after you disturb them. Apply the poison, and leave. Check the next day. The insects will either be dead, or not on alert looking for a threat anymore. If necessary reapply.
If the nest is located within a structure, search for products that work in such a situation. The home store will carry them. Instead of a spray the poison takes the form of bait. The insects carry the poison into the colony. Apply as indicated.
5) Option: Call the exterminator
If the store bought poisons don't work or the colony is too large, you can either buy a bee suit, or call the exterminator. The bee suit really only allows you to handle insects for a much longer time without having to worry about getting attacked. I don't think it will help the average home owner. The methods remain the same, except that you now can physically destroy large nests.
The exterminator uses pretty much the same methods detailed in the step above, except that he is so good at applying them that they work every time. Know when you are beat!
Small colonies do not always have to be removed. In some cases they can be approached safely. Plenty of products are available at home centers. Always approach a nest with a plan for how to get away as fast as possible. Always stay clear of killer bees. These ladies are dangerous.