To a great extent, the animals who rebel do so out of opposition to Napoleon's actions as leader. There are some areas where there are differences as to why they decide to rebel or openly question his actions, but most of it boils down to dislike of Napoleon as leader. For example, three hens consider it murder that Napoleon demands that they surrender their eggs in order to fulfill the arrangements he has made with Whymper. In destroying their eggs, this represents an act of rebellion. When Napoleon wants to end Sunday meetings, some porkers disagree. In their dissent, another act of "rebellion" is construed. When Boxer questions Snowball's retelling of history in suggesting that Snowball was actually treasonous to the ends of Animal Farm, it is seen as an act of rebellion. This becomes broadened to any animal that engages in a collaboration of any kind with Snowball. The cause of these associations are the belief that Snowball could be a better leader and possesses better ideas than Napoleon. In the end, the dissatisfaction with Napoleon in the fundamental cause of rebellion, its perceived end and its actual result. This is something that in either form Napoleon strikes down with intensity and force, as seen in chapter seven when Napoleon coordinates public confessions and demonstrates public execution in the most brutal of fashion.
The animals first rebel against the farmer, Mr. Jones. This most likely happens because Old Major, the pig, gathered them and spoke of his dream of an animal-run eutopia. After this rousing speech (and the old aged death of Old Major), Jones gets extremely drunk and his animals go hungry the next day as he is sleeping it off. The animals get into their feed (a mini-rebellion), and when Jones responds violently to this, the animals rally and drive him from the farm. Snowball and Napoleon then take over the running of the farm. Some animals do rebel against Napoleon later (hens push their eggs off the ledge rather than give the eggs to Napoleon), but the major rebellion is against Jones. The reader can wonder if the rebellion would've happened at all without Old Major planting ideas of eutopia in the animals' minds. They had experienced cruelty and deprivation from Mr. Jones before, and it is probably this coupled with Old Major's words that led the animals to revolt against their human master and try to run things for themselves.