Grass-Fed/Grass-Finished: Pretty much all cattle raised for meat feeds on grass soon after weaning. But most are transitioned to grain, which facilitates faster weight gain and is believed to result in more marbling. Cattle that continue to feed on grass in pastures for the duration of their lives are called grass-finished (though grass-fed is also common).
Grass-finished beef is more environmentally friendly than grain; pasture-raised animals often play a key role in a sustainable farming system. As far as your plate goes, grass-finished beef tends to be leaner than grain-fed, and the flavors of the meat vary according to the geography and the types of grass the animal ate; this is considered a desirable feature for fans.
Many restaurants serve grass-finished beef from Australia. It’s perfectly fine, but don’t kid yourself that you’re doing the planet any favors by eating meat that’s been shipped from halfway around the world.
Natural: When it comes to meat, this means it contains no preservatives or artificial ingredients that fundamentally alter the product. Meat from animals given hormones and/or antibiotics can still be labeled as "natural."
Naturally Raised: This is a specific certification program in which animals are raised without hormones or antibiotics, ever. Note the difference in language between "natural" or "all natural" and "certified naturally raised."
Dry-aged vs. wet-aged: Aging meat improves the taste and texture no matter how you do it. The easiest and most economical way restaurants and producers do this is through wet aging, in which cuts are kept in vacuum-packed plastic for a few days. Dry-aging, on the other hand, entails hanging whole sides of beef uncovered in near-freezing temperatures for several weeks.
Dry-aging results in more concentrated flavor and ultra-tender meat; but moisture loss from the process results in an overall loss of volume and a more expensive steak. Wet-aging lets you retain all of the volume, but you won’t get the same depth of flavor. Choose dry-aged if you’re adventurous, up for the splurge, and know you’ll appreciate the intense flavor.