To put it simply, an organic produce are food or products that are free from the use of pesticides, chemicals, synthetic fertilizers, sewage sludge, genetically modified organisms, or ionizing radiation.

The USDA National Organic Program (NOP) defines organic as follows:

Organic food is produced by farmers who emphasize the use of renewable resources and the conservation of soil and water to enhance environmental quality for future generations. Organic meat, poultry, eggs, and dairy products come from animals that are given no antibiotics or growth hormones. Organic food is produced without using most conventional pesticides; fertilizers made with synthetic ingredients or sewage sludge; bioengineering; or ionizing radiation. Before a product can be labeled “organic,” a Government-approved certifier inspects the farm where the food is grown to make sure the farmer is following all the rules necessary to meet USDA organic standards. Companies that handle or process organic food before it gets to your local supermarket or restaurant must be certified, too.

This is the definition from Wikipedia about natural product:

A natural product is a chemical compound or substance produced by a living organism—that is, found in nature.^[2][3] In the broadest sense, natural products include any substance produced by life.^[4][5] Natural products can also be prepared by chemical synthesis (both semisynthesis and total synthesis) and have played a central role in the development of the field of organic chemistry by providing challenging synthetic targets. The term natural product has also been extended for commercial purposes to refer to cosmetics, dietary supplements, and foods produced from natural sources without added artificial ingredients.