If you're like me, you scratch your head a bit in the egg section of the supermarket. Especially if you know that in the case of other foods certain designations, like "all-natural" actually have no set meaning, and certain standards, like USDA Organic, aren't particularly stringent. A recent issue of Consumer Reports lays out the definitions of the egg terms you might see. I summarize and add unsolicited comment.
Brown vs. White- A difference in the chickens' breed produces the color change. There's no difference in taste. Brown-egg-laying chickens eat more which means brown eggs are more expensive.
Organic- A USDA standard set on use of pesticides and other chemicals in the chickens' feed. I would suggest as with all organic products you look for third-party certification.
Nutrient enhanced- Chickens are fed a diet higher in Omega-3s, vitamin E, and/or lutein. I don't know how much this helps but I would guess not much. The foods in which these nutrients occur naturally are going to be your best bet.
Free-range- Chickens spend some amount of time outdoors. Please note that the USDA, that guardian of our morals, sets no lower limit. Buy from a farm you know if humanly possible.
Cage-free- Chickens "permitted to roam in barns but not outside." Completely unregulated category.
Pasture-raised- Chickens kept in pens in a pasture, where presumably they feed on grass and bugs, aka natural chicken food. Pens are rotated, according to CR.
Pasteurized- Thanks to savvy Frenchman Louis Pasteur, about whom I had a book when I was a child (I had any number of books about inspiring figures; the one that stuck was about Martina Navratilova), we now no longer have to fear food poisoning from certain raw farm products. Pasteurized eggs are warmed in the shell and then the shell is waxed. Apparently they are used in "hospitals and nursing homes" and also in recipes (beef tartar comes to mind) which call for raw eggs.