These are kinda old-fashioned terms, but it's a fun exercise, so I'll play--
Alpha. Calmly aware of their status at the top of the pecking order. Do not take power, but have it willingly given to them by other pack members. Dogs seek them out excitedly, but they choose whom they wish to play with. Rarely need to discipline with more than a hard stare and a low growl, but will fight if necessary. Typically look up to the human as the pack leader, although may test from time to time.
Beta. Insecure middle-rank dogs. Always trying to move up in the pecking order, these are the dogs who are most likely to provoke fights. They are the dominance humpers.
Omega. I would divide these guys into two categories: confident submissives and fearful submissives. Theo is a confident submissive. Happy to be a follower, they seek out the alphas by wiggling, face licking, and offering bellies--they are delighted if the alphas play with them but take it OK if they choose not to. Frequently humped by "beta" dogs, but try to avoid fights by turning humping into a game. The fearful submissives may run away from a provocation, or may fear bark or even fight back. They frequently find their tails between their legs at the dog park.
One point: None of these terms is static. Recent research on wolf and dog packs (don't have cites at hand, sorry) shows that status hierarchy is fluid and changes according the composition of the pack and the situation.
I got to thinking about this when I was watching a show on wolf packs one night..... and then trying to classify what I thought some of my past dogs would have been considered.* (as well as my present one)
My last Lab had what I would call the ideal Lab temperament, but when I read Nathan's descriptions I'm not sure that any of the three personalities accurately described her.* She was I guess what Cesar Millan would call very "balanced."* Extremely gentle and calm, would never "start" a fight in a million years, but she wasn't submissive or at all nervous with other dogs.* In fourteen years she only had one bad encounter with another dog and "it" attacked her.* She did submit, but then she was only a young pup at the time.* She was a very confident dog, but at the same time never showed any aggression or real dominance towards anyone or anything.*
My current pup (a year and a half) is very excitable around other dogs.* She gets along fine with the easy going dogs, but often seems to get a negative reaction from those who aren't so easy going.* She's never been in the position of actually having another dog seriously turn on her so I don't really know what her reaction would be.* I'd have to say that she tries too hard with other dogs and comes on too strong.....but she wants to play not fight.* Her doggie manners do seem to be improving somewhat.* She's still very puppy in many ways, so things could change.* She is a very different personality from my last Lab in so many ways, but I'm not sure which I would classify her as either.* It doesn't seem that clear cut to me.