The first conflict seen is between the animals and Mr. Jones. The animals are sick of being treated poorly by him. The drinking problem he has developed is the cause of this and the terrible job he is doing running the farm but he has control over all the animals. Mr. Jones takes all the product of their labor but gives nothing of value to the animals in return. This conflict is resolved when the animals revolt and force Jones and his workers off the farm. The animals are now free to run the place and make it the utopia they dreamed about.
The second conflict is between Snowball and Napoleon. They start out working together to formulate the principles of the Animalism philosophy. They work together to paint the seven commandments on the side of the barn. As time goes on however, Snowball and Napoleon argue over the future of the farm and how things are run. Snowball is intelligent and very passionate and eloquent in his speeches to the animals. He truly believes that a utopia is possible and that all the animals are equals. Snowball works hard to spread Animalism and to improve the Animal Farm. He wins the loyalty of the other animals. However, Napoleon is sneakier. He manages to get the other animals on his side between meetings. Napoleon, from the beginning of the uprising, has cheated the other animals. He never really makes any contributions to the Rebellion, the formulation of Animalism, or the establishment of the new society at the Animal Farm. He doesn't work; he just supervises. He keeps the apples and milk for himself and the other pigs. Napoleon tires of the competition with Snowball so he sets his trained guard dogs on him and forces him off Animal Farm.
As time goes on, bad things start to happen on the farm. Napoleon always blames it on Snowball. Napoleon says that Snowball sneaks onto the farm at night and sabotages their work. Napoleon goes on to accuse animals on the farm of conspiring with Snowball to destroy things. He has his dogs attack and kill four pigs and numerous other animals. All the hens who had objected to selling their eggs were killed. Napoleon is corrupt and has seized all the power for himself. Napoleon finally has Snowball killed.