One key point about the rebellion is that it is motivated in large part by Old Major's speech, given shortly before he dies, in Chapter One. Old Major demonstrates to the animals that they are being brutally exploited, and that Man is their enemy. If Man were out of the way, then animals could live happy and productive lives. This speech, intended to echo the writings of real-life philosopher Karl Marx, is in many ways the catalyst for the rebellion and all that follows.
A second key point is that even though almost all the animals participate in the uprising that drives Jones from the farm, the pigs (seen as the most intelligent) are the leaders. Among them are Napoleon, a fearsome looking boar who is a natural leader, Snowball, an energetic, brave, and idealistic pig, and Squealer, who will emerge as a sort of propaganda officer. These three lead the successful rebellion, and it emerges that the pigs will get some benefits from their leadership (like, for example, apples in their mash) that the other animals do not receive. We also learn of a power struggle between Napoleon and Snowball that sees the latter driven from the farm.
Finally, another important point is that the rebellion, while conducted with an almost utopian idealism, eventually flounders as Napoleon emerges as a leader as brutal as Jones was. This, in fact, is the key point of the entire book: that power must be checked in order to preserve basic liberties.