The most important thing to understand when analysing Animal Farm by George Orwell is that the entire novel is intended as an allegory of the Russian Revolution, the formation of the Soviet Union, and Josef Stalin’s rise to power. Therefore, almost all characters and events are intended to directly symbolise something similar from real life.
A main theme is therefore the Russian Revolution, but the novel’s scope is not completely limited to this event. More broadly, it is also concerned with the general theme of violent revolutions. There is also a related sub-theme focused on the power of state propaganda, as we frequently zoom in on Squealer’s various efforts to distort or hide the truth about events.
Because of these more general themes, the story’s message is not limited to criticism of Josef Stalin and Soviet Russia, but is also criticism directed at revolutions in general. The novel warns that people can easily end up in a post-rebellion situation that is just as bad as before - or perhaps even worse - especially if they are also vulnerable to state propaganda. In particular, Orwell points to the way in which a revolution can create a power vacuum, and those who are able and willing to fill it may often be worse than those they are replacing. The novel also acts as a warning that although the principles of Communism appear sound, they are unlikely to work, and that in the end Communist societies are likely to look exactly the same as Capitalist ones.
You can find much more advice about interpretation in the following sections!