(please excuse my terrible drawing!)
Modern
This is what we would call "modern" style of chicken raising. You've got many birds crammed in tiny cages where they can't move. Their beaks are clipped short to prevent canabalism. This allows large farms to produce a great amount of chicken meat in a short time, and it is most likely this is the way your grocery store eggs were layed.
A miserable existence.
Free Run
Ah free-run the new buzz word! Just enough "free" in there to make you feel less guilty! Free run are raised in a barn and don't get to see the sunshine either. Their breaks are also very likely clipped short.
Much nicer then the modern method at least.
Free Range
Free range means the chickies get to bathe in the sunshine, and probably eat some grass & bugs!
Traditional coops & runs work as free range, that is how we have our layers. Keeps them safe from predators but allows them as much exercise as they want.
Tractor! Also counts as free range! They are out in the sunshine, eating all the grass and bugs they want, digging in the dirt being chickens.
You should never have to clip beaks in any free-range method.
Truely free-range, no fences to keep them in... or predators out.
Well, there goes dinner!
There are two other relevant terms here.
Organic is a process of certification, there are steep dues to pay and lots of rules to follow. Not really attainable for most small farmers.
(Although there are some weird rules about producing small amount you can still call them organic... I'm not sure)
Pesticide-free, no-hormones, etc. Mean exactly what they sound like; but beware, big companies like to use things that are illegal anyway to promote their products! It's great to call your milk hormone free, but you can't give hormones to dairy cows in Canada anyway!