In the world of food, "cultured" essentially means fermented—the chemical process of breaking a complicated substance down into simpler parts, usually with the help of bacteria, yeasts, or fungi. Different species of the bacteria Aspergillus, for example, are often used to culture barley or soybeans in order to produce miso or to culture rice for the creation of amasake.
Fermented foods are far more common than we realize; for example, yogurt is the most commonly eaten fermented food in the United States. Because certain methods of fermenting foods with certain types of bacteria result in the production of lactic acid, you might sometimes hear the term "lacto-fermentation" used to describe one particular type of food culturing.
Sauerkraut—another name for fermented cabbage—is an example of a cultured vegetable. An Asian version of fermented cabbage, especially popular in Korea, is called kim chee. A basic fermentation process is used to culture the cabbage in kim chee, and chili pepper, garlic, and salt are typically added to give this cultured vegetable its pungent character.
During the fermentation process, bacteria both enrich and preserve the food. This process breaks down elements that are sometimes difficult to digest, such as gluten and sugar. The activities of these bacteria benefit digestion, increase the availability of vitamins, and promote the growth of healthy flora (bacterial balance) throughout the digestive tract. Because of their helpfulness in the fermentation process, and because they usually help to counterbalance the presence of other potentially problematic bacteria in our digestive tract, the bacteria deliberately used to help carry out fermentation are often referred to as "friendly bacteria."
Because bacteria and other microorganisms exist everywhere in our environment, and because all of the foods we eat can serve as food for bacteria as well, most foods—including vegetables—will naturally ferment. Some websites encourage the use of natural fermentation as a way to increase the value of vegetables. In this context, "natural fermentation" means leaving the vegetables in a container at room temperature for four to seven days. No additional bacteria are deliberately added to the vegetables, and no heating process is used.
While it's easy to imagine how naturally cultured, raw vegetables might have greater bacterial populations than commercially fermented vegetables (like sauerkraut you buy in the grocery store)—and while raw foods can also contain enzymes that may have activity in our digestive tract while the foods are being digested—I don't recommend culturing your own vegetables in this way. From my perspective, there are too many potential health risks involved when you simply cannot know what microorganisms are at work altering the vegetables. "Friendly" bacteria—like Lactobaccili—may or may not be present. The same goes for "unfriendly" bacteria that might cause illness.
I like the idea of cultured foods and believe they can easily be as nutrient-rich as non-cultured foods, as well as potentially easier to digest. However, I recommend consumption of cultured vegetables and other fermented foods that have been cultured under controlled conditions where safety was monitored and specific bacteria and fungi were used.