How does Animal Farm break down into the typical parts of story structure?
Old Major's speech is the Exposition: it gives the information necessary to understand the events that follow.
The Rebellion and following events with the farm are the Rising Action: each event moves the plot closer to the climax, with the characters facing adversity and trying to overcome obstacles. This is the longest part of the novel, as each event dovetails into the next, and each decision informs the actions of the next part.
Napoleon's adoption of human habits is the Climax:
...out came Napoleon himself, majestically upright, casting haughty glances from side to side, and with his dogs gambolling round him.
He carried a whip in his trotter.
(Orwell, Animal Farm, msxnet.org)
Although all the hints were there, this is the first time it is obvious what Napoleon's intentions were from the start; all his power-grabs and brutal actions come together to show how Napoleon was becoming less "animal" and more "human."
The events at the poker game are the Falling Action/Resolution: the animals see that the pigs never intended to have an equal society, but instead have set themselves up as surrogate humans, and now will work the animals even harder than Jones did. At the end, they cannot tell the difference between the pigs and the humans, cementing Napoleon's success in taking total control of the farm.
George Orwell uses the fable, Animal Farm to show the basic nature of all government systems. The farms represent countries having different forms of governments and the animals represent the people. It is made of incidents which are all interrelated. It has an exposition in which the revolutionary ideas are presented by a boar representing Karl Marx. A successful revolt based on the exposition results in unseen complications when two hostile leaders(represented by pigs) emerge. The point of conflict arises both between the two antagonistic leaders and the clash of interests between the commoners and the representatives. The key moment is the point where the people realise that there was no point to the revolution as its reality was only the change of power and their state had only disintegrated further. The climax is reached as a chain reaction to the preceding events and arrives at the point when the old horse finds it difficult to differentiate between the faces of the humans and their representatives, the pigs. Thus the climax enforces the basic similarity of all power groups and exposes the false idealism of the exposition.