USDA Certified Organic does have a legal meaning that requires certification in order to have the logo on the packaging. Some other states and trade groups had standards prior to that (on edit - I think Oregon Tilth was/is one?). Words like "natural" and "green" mean nothing.
"organic" is an unfortunate word because it can mean "natural" when used in a literary way (e.g., arising organically in a discussion), and "organic chemistry" is chemistry of hydrocarbons including everything from biochemistry to petroleum.
For agriculture, in short, it means the crop was grown without synthetic (man-made) pesticides (doesn't mean no pesticides), and without synthetic (man-made, e.g., ammonium nitrate pellet) fertilizers.
Michael Pollan's book The Omnivore's Dilemma is a great read (I plowed through it in a few days) and he researches a lot about modern agriculture including "big organic". A very small number of companies (Cascadian, Organic Valley, etc.) control a big share of organic production in the U.S.
As much as I feel like I can afford it, I try to buy organic or IPM or local. It's hard to find a combo of all 3. I worry more about it on fruits/veggies with thin skins, etc.
(IPM = integrated pest management, which works toward minimal use of pesticides, using less-toxic compounds, prevention, natural predators to control insects, etc. Most but not all organic uses IPM. (on edit, IPM does overlap with conventional (non-organic agriculture - how I hate that "conventional" = pesticides when my grandparents sure didn't use pesticides on their crops) IPM doesn't really have a branding scheme though because it's a system of doing things, not really standards, though Wegman's labels some of their produce IPM.)
USDA info -
http://www.ams.usda.gov/AMSv1.0/getfile?dDocName=STELDEV3004446&acct=nopgeninfohttp://www.ams.usda.gov/AMSv1.0/getfile?dDocName=STELDEV3004443&acct=nopgeninfo