Cage free, free range, pasture raised and organic; as meat consumers demand more information about the way their food is raised, producers have come up with a cornucopia of terms to try and pass that information along.
Organic is a label regulated by the USDA according to very strict rules that prohibit giving animals antibiotics and growth hormones, as well as restricting the feed they are given. Genetically modified grains are off the table, as well as any plants treated with persistent pesticides or fertilizers outside of a small approved list.
Grass-fed vs. Pasture raised - Not strictly regulated, pasture raised animals can be fed grains such as corn as long as they are not raised in confined spaces. Grass-fed animals are generally raised on nothing but grasses and hay from plants like alfalfa for the greater portion of the animal's life. These claims don't necessarily address antibiotic or growth hormone usage.
Cage-free vs. Free range - Usually applying to poultry, free-range ensures animals have outdoor access as well as being free from confinement. While there is no set definition for cage-free, freedom to practice natural behaviors is implied; outdoor access may not be.
American Humane Certified - This certification must be approved by third-party auditors according to the American Humane Association's specific standards. Different species have different specifications, but the use of added growth hormones is prohibited, as well as non-therapeutic or sub-theraputic use of antibiotics.
Source verified meat has simply gone through processes guaranteeing where the meat came from, making it easier to track the source of any potential problems.
Natural products have no artificial ingredients and are minimally processed. This label doesn't specifically address whether an animal was free-range or pasture raised, nor does it guarantee the animal was antibiotic free.
Hormone free labels mean just that an animal wasn't administered growth hormones to shorten the time before it is "finished" for the processor. Hormones are prohibited in raising poultry and pork in the United States, but are common in conventional beef production.
Lean meat claims are regulated by the FDA as containing "less than 10g total fat, 4.5g or less saturated fat, and less than 95mg cholesterol" per 3.5 oz serving. The rules for "extra lean" lower the allowable total fat to 5g and saturated fat to 4g or less.