A true organic gardener uses a method known as integrated pest management (IPM) to control insects. This may mean importing other insects to eliminate certain kinds of pests. For example, cabbages suffer from cabbage aphid, but Diaeretiella rapae a tiny wasp lays its eggs inside the aphid and the hatching eggs kill the aphid. This control may take a week or two, but is very effective. Similarly, bugs in your lawn can be controlled by Milky Spore disease. In both cases the killer is organic and stays around until it is needed again. One treatment with Milky Spore disease, for example, can last up to 15 years.
Using beneficial insects to control pests is difficult because the insects usually fly away after they have done their job, and sometimes before they've done their job. However, you can attract beneficial insects by planting flowers that attract insects. Queen Anne's Lace, yarrow, clover, alyssum, and many herbs attract beneficial insects to your garden.
Other plants are used as insecticides; pyrethrins, which come from the chrysanthemum plant and are best sprayed for soft bodied insects. Neem is made from the Neem tree (from India) and is a good organic pesticide. It is best for insects that chew on your vegetables. If the insect does not feed on the plant it is not harmed by Neem. Neem is also used in soaps and creams for skin care. Rotenone is made by synthesizing certain plants. It breaks down in sunlight in about six to eight days. It is moderately toxic to humans and should not be inhaled or used without a mask. Rotenone was removed from the organic approved list in 2005.
You should note that because they are organic, doesn't mean that you can spray them without protection. When using any pesticide be sure to wear a mask or respirator. The ones mentioned here are organic, but they are concentrated and you will need to wash your hands after using them and keep pets away until the spray has dried.
There are other pesticides made from bacteria. Any insecticide with Spinosad or Bacillus thuringiensis are considered organic. (BT is also known by the trade name Dipel or Thuricide) They are best used on plants where the insect will ingest the bacteria. I also use Safer's insecticidal soap. It appears to be reasonably safe and kills aphids and other crawling bugs. You can get all of them from either Jamestown Hardware or Secret Garden. As an aside, if you have aphids on your roses, simply blast them off with a jet of water rather than using a pesticide.
Another class of pesticides are made from naturally occurring substances. Diatomaceous earth is made from the remains of diatoms. It is ground up and can be dusted around plants to keep slugs and snails off them. Each dust like particle has sharp edges that cuts the monopod of crawling bugs and causes them to dehydrate.
Then we come to keeping bigger pests off your garden. Around here keeping deer out is a big problem. I've found that Liquid Fence or Deer-Off work well. But if you are going to spray it on blooms such as tulips - which deer seem to like a lot - it ruins the smell. Don't use either product on food plants. The base of the product is rotten egg solids and garlic, so it's not going add a savory flavor to your dinner.
Companion planting is another way to keep animals and insects from your plants. Planting marigolds around your lettuce and salad greens keeps rabbits off them. Planting garlic next to roses keeps aphids from the roses. The smell of the plant is masked by the companion plant and the insects stay away.
The best gardeners are watchful gardeners. A morning walk through the garden might reveal holes in a leaf that weren't there before. First, you need to find out what caused the holes. Tiny holes that look like a shotgun blasted the leaf might be caused by flea beetles. Treatment after the fact is possible by spraying the leaf with Neem. However, a better method is to cover the plant row with spun fleece and prevent the arrival of the beetle.
Similarly you might find larger holes. These might be caused by caterpillars of the cabbage white butterfly or by earwigs. Covering the plants with spun fleece prevents the butterfly from laying eggs, and spraying with rotenone helps kill the caterpillars. For earwigs spray Spinosad. If you find caterpillars, the easiest control is to pick them and crush them either by dropping them on the ground and stepping on them or squeezing between your fingers.
The use of a mulch on the soil also helps to prevent soil dwelling insects from climbing onto your plants and spun fleece (sometimes called Reemay) prevents flying insects from landing on your plants.
There are then, many methods of organic control. You should identify the best ones for your situation and use them. Your plants may not be as bright and shiny as those in the stores, but you will be sure that they have not been sprayed with insecticides.