They're incredible, they're edible, they're confusing.
Brown, white, cage free, organic, omega 3: What does it all mean?
Brown vs White
There is no difference between these two types of eggs. Chicken eggs actually come in a variety of colors including blue and green, along with the conventional brown and white ones. The color of the egg is based solely on the genetics of the chicken. There is zero nutritional benefit to going brown over white.
Organic vs Non-Organic
If an egg is certified organic, then the feed that was given to the chicken was organic. Also, organic eggs come from chickens not exposed to antibiotics. Chickens that are organic are not necessarily considered cage free or free range.
Cage Free vs Free Range
Cage Free isn't much different than regular eggs. Although they aren't raised in battery cages, they are still confined to a very small space. Don't be fooled, these guys still have their beaks and wings clipped and can still contain antibiotics and other exogenous drugs.
However, free range chickens are given exposure to the outdoors for a portion of their life. That's about the only standard for it. Other than that, free range eggs can come from chickens with almost the same living conditions and confinement of cage free and regular eggs.
These are the highest grade eggs you can find. Pasture raised chickens are free to roam in an open field and eat an organic diet. Thereby, raising the amount of omega 3 fats and micronutrients including vitamin A and E, according to a study conducted at Penn State. They also aren't given antibiotics or other drugs or hormones.
Battery Caged vs Free Range Chickens
These are kind of redundant in my opinion. Chickens who are raised vegetarian are still going to eat insects. I see no point in ever spending the extra money to buy these.
This only means that the chickens were fed diets containing flax seed, thus raising the amount of omega 3 fats in the yolk. The difference is pretty negligible though. You'd be better off getting your omega 3s from quality sources like salmon, chia seed, or avocados.
So, choose whichever type of egg fits your budget, but don't waste your money on brown eggs. Ever.