What happened between seasons 2 and 3 of Mad Men?

The people from PPL have chosen Duck as the boss of the new agency. When they met at the end of episode 13 (from Season 2), they tell Don that Duck will be his new boss. The people from PPL know that Don is the genius mastermind behind everything and without him, the agency is worth nothing. Don is aware of this and tells the people from PPL that he will quit as soon as Duck will be the new boss. This is his right since he doesn't have a contract, so he cannot be forced to work with them (In the beginning of Season 1, Roger gives Don more money but without having a contract).

The main scene leading to the events you described reads is here:

(Don) I think it sounds like a great agency, and I think Duck is the man to run it. I just don't think I'll be a part of it.

[...]

(Duck) You can either honor your contract or walk out that door with nothing and start selling insurance.

(Don) I don't have a contract.

(Roger) We're close.

(Don) We didn't think we needed one.

[...]

(Don leaves the room; Duck is mad about Don; Cooper looks very angry at Duck)

(Man from PPL) Duck, would you excuse us for a minute?

If they really wanted him to become the new director they would not ask him to leave the room after such a scene. Off screen they would have decided about firing Duck and keeping Don in order not to lose money and get a better agency.

Duck was a very self confident man, with not a lot of real talent to back it up. He was an alcoholic who lost his wife in divorce. His beloved dog was returned to him by his ex-wife when the dog was no longer welcomed by her next future second husband. Duck proclaimed love for the dog and when he decided he did not need him, he simply took off the leash and released him onto the streets of NYC. It was at that moment that you knew that he had no heart. When Roger told him that he would not be a partner, especially after the loss of the American Airline account, he thought he would encourage the merger as a way to ensure his promotion as president. He also thought it would be away for him to then destroy Don. He was once again, dead wrong, and took himself out of the game by revealing his hatred of Don, instead of his loyalty. Sort of like the story of the dog who looks in the pond to see aother dog with a bigger bone. When he goes to snatch the bone, he drops the one that he is holding, since it is actually the dogs own reflection that he sees in the water.