In all our years of gardening, there is one thing we know for sure--pests can find their way into our gardens. Experience has given us a few organic means of combatting these 
pests. (for tips on prevention methods see: Pests: Prevention Methods).
Some general tips to keep in mind:
- Controlling pest's eggs will be easier and more productive than waiting until pests are mobile.
- Acting early can mean your efforts will reduce insecticide use.
- Apply sprays that work on contact carefully, not broadly, and in the morning or evening when many insects are less active, allowing you to better target pests, while cooler temperatures often allow the product to work longer.
- Scout for pests regularly.
Ready-Made Organic Products and Concentrates
Although these products are organic, they should be used with care and caution. Always follow the manufacturer's instructions on which pest and plants it is useful for, how to apply, when to reapply, and any other guidelines. Some products cause pests to die immediately and others cause them to die over time; it helps to know what to expect.
Bacillus thuringiensis (Bt) and B. thuringiensis var kurstaki are bacteria that kill caterpillar pests by disrupting their digestive system. It may take a few days for caterpillars to die, so don't expect immediate results. Repeat applications may be necessary as new generations hatch. Bt is helpful against tomato hornworm, cabbage loopers, cabbage worms, corn earworm (if mixed with vegetable oil and applied to silks), and cutworms (if mixed with moist bran and molasses and scattered over surface of beds and other caterpillars). Bt comes in liquid or dry forms.
Copper compounds are useful for controlling certain types of fungus like mildews, anthracnose, blight, and more. Many formulations are organically approved. See packaging for specifics.
Diatomaceous earth is made up of the fossilized remains (powder) of 
diatoms (tiny aquatic organisms) that have turned to silica. To soft-bodied insects--like slugs, aphids, fleas, mites, and ants--diatomaceous earth is actually very sharp, causing them to dehydrate when they come in contact with it. Diatomaceous earth can be applied to plants and on the soil around plants to create a protective barrier that deters and kills these soft-bodied pests.
Insecticidal soap is a good complementary treatment to Bt or can be used on its own as a first line of defense. Insecticidal soap is a liquid spray that works best in direct contact with susceptible soft-bodied pests and insect eggs. Because it is effective only upon contact and washes off readily, there aren't ongoing impacts to pests and beneficial insects after the spray has dried. Some insecticidal soaps are also labeled for use against powdery mildew on some plants.
Kaolin clay-based products are liquid or dust that can be mixed into a liquid and employed to prevent insect damage by creating a physical barrier between the pest and the plant. It is also used to protect plants from sunburn and help to reduce heat stress in plants. It is often effective against aphids and squash bugs.
Neem is an extract of the Indian Azadirachta indica tree's seeds and leaves. It acts by smothering to prevent feeding and breeding. Results may take several days, and it will likely need to be reapplied. Neem is effective against many eggs, nymphs, and adult insect pests like aphids, mites, scale, leaf hoppers, white flies, caterpillars, mites, mealybugs, thrips, and lawn grubs, including Japanese beetles. Neem is also effective against fungus and bacteria and can be found in liquid or dry forms.
Seaweed (kelp) extract is packed with micronutrients and minerals along with some macronutrients. It can be used to boost plant health and boost resistance to pests, diseases, and stresses like drought or pest damage. It has been recommended as a foliage spray to help combat blossom end rot; however, note that calcium deficiency, which causes blossom end rot, is quite often caused by inconsistent moisture. It can be found in a liquid or dry form; generally, the liquid form is better for a sprayer, as it is less likely to clog, and the powder can be incorporated into the soil.
Spinosad is a soil bacterium in dry or liquid form that can be toxic to certain insects. Insects will not die immediately, so exercise patience and reapply as per product recommendation.
Sulfur is another organically approved fungicide that can be used to combat fungal diseases like powdery mildew or blight.
Homemade/DIY Solutions
When using homemade sprays, first test a small leaf area and wait 24 to 48 hours to observe if it causes any damage before applying further.
Compost tea is praised for pest and disease prevention when applied as a root drench or fine foliage spray. Some studies show some benefits, mostly concerning specific crops and specific disease. If you plan to use a sprayer, place aged compost in a cloth bag to contain the solids and prevent your sprayer from clogging. Place the bag in a large bucket, and using one-part compost to five parts water, steep it in a warm place for 1 to 2 weeks, stirring regularly. Then remove the bag and put the liquid in a sprayer or watering can.
Garlic and hot pepper spray is used for a wide assortment of garden pests including aphids, leaf hoppers, squirrels, caterpillars, birds, deer, vampires, and more. Use a 
blender to combine several chile peppers (fresh or dried), about a dozen garlic cloves, and about 2 cups of water. Let the mixture sit overnight and then strain in the morning so it won't clog your sprayer. Add about 2 tablespoons of non-toxic, biodegradable soap to the concentrate. When ready to spray, add a tablespoon of the concentrate to 16 ounces of water and put in the sprayer.
Spraying in the morning or evening helps to protect pollinators; less of the spray evaporates, and it helps ensure plants are not stressed or at risk of burning from the spray. Reapply after a rain and every 7 to 10 days as needed.
Milk spray This treatment works great for powdery mildew. Dilute milk to 20-50% in water and add a couple of drops of non-toxic, biodegradable soap. The soap helps the spray stick to leaves, rather than rolling off. Apply weekly, spraying plants thoroughly for good coverage. Like many powdery mildew treatments, it is best to start this as a preventative action before plants are infected.
Soil solarization uses plastic sheeting to heat the top of the soil during hot summer months, which at high enough temperatures can kill many soil organisms, including those that cause disease, nematodes, weed seeds, and seedlings. One drawback to this method is that it will also kill beneficial organisms, so use with caution.
Wood ashes can be used around plants to deter slugs and snails, which avoid crossing it. It will need to be reapplied after rain or watering. Wood ash does raise soil pH, making soil more alkaline, so be cautious using it if your soil is already alkaline.