It sounds like you might mean simply "organic", rather than "organic compounds". trophyhunter answered your question as written--organic compounds are indeed those that contain carbon, and Organic Chemistry is an entire course on just those compounds. The other use of the term "organic" is supposed to mean, generally, earth-friendly, or not using artificial chemicals in food, agriculture, or manufacturing in general. The legal and general use of the term are different, and has evolved as food producers have used misleading advertising to make their products seem healthier than they are. It is only in the last hundred years or so that this has even been necessary--before artificial additives and fertilizers, everything was already something that was found in nature.
Benefits are that when we eat fewer additives, we may decrease our risk of diseases such as cancer; we don't really know what long-term use of some of these chemicals can do, both to individuals, humans, and the earth overall. Problems arise when people don't really understand what the term means, and reject everything that is not "natural"; getting diseases is "natural", for example, but we don't want to go back to getting smallpox!
I've attached a couple of links--this is a complicated subject, with long-term effects we don't really know, yet.