As I understand it regular eggs are from hens caged in tight quarters and not let outside. Free run eggs mean they are allowed to roam in crowded indoor pens, and free range means access to an outdoor area and the ability to scratch around hunting for food like a real chicken. (And real chickens DO eat bugs.)
I used to live near a factory-type egg farm and the hens upper beaks were clipped so they were only able to eat out of dishes. I bought some 'used' hens and had the darndest time chasing them outside at first. They eventually figured it out but still weren't able to eat like normal hens. In the factory farm they were subjected to artificial light making two days out of one so they would lay more eggs more quickly. Prison camp.
For years I have bought local egg farm eggs figuring that at least they were relatively fresh, but this year I switched to genuine certified free range eggs, and there is a world of difference in appearance and flavour. They are worth the extra cost.