In "Animal Farm", how does Napoleon influence the others animal in their decision making?In Orwell's allegory 'Animal Farm.'
Napoleon first uses repetition of slogans in public chants or incantations to brainswash the animals, particularly the sheep. Here a type of "group hysteria" and "bandwagon" technique gets the animals go 'go with the flow' in the direction he desires. You see this particularly in the change of the chant "Four legs good, two legs bad" into "Four legs good, two legs better" where the animals don't even seem to recognize the difference. Unlike Snowball's attempts to educate the animals through his multiple social programs and committees, Napoleon rather exploits the animals by keeping them in their ignorance.
Just as important is Squealer's role as middle man, public spokesman and propaganda expert. Napoleon is not particularly gifted at public speaking, but at least he has enough sense to confer this role to somebody else more influential and persuasive. Aptly stated was the observation that Squealer could turn white into black and black into white. His help is crucial in establishing Napoleon's influence over the other farm animals.
Napoleon also uses reward and punishment to condition the animals' responses. They are given extra portions of food (rarely but sometimes) after battles or during celebrations. The old crow Moses' tales of Sugarcandy Mountain in the hereafter and the more immediate prospect of the benefits of the windmill to make life easier on the farm use the "carrot before the nose" tactic of promise of a better day forthcoming. Here both religion and philosophy are truly "the opiate of the people" (or "animals," as this is an allegory). On the other hand, the animals are severely punished for any form of resistance or insurrection, such as the hens' rebellion over having their eggs confiscated. With the help of his secret police (Bluebell's puppies, trained to kill and now adult), he extorts forced confessions and even has a couple of animals executed in consequence of their "crimes."
By living in the house with the pigs, Napoleon creates a physical division and elitist privileged class whose members are at his beck and call to curry any favour they can. The establishment of such an oligarchy is indipensible in Napoleon's keeping everything under control, for without the help of these privileged few, Napoleon would be powerless.
Napoleon also eliminates Snowball as his rival by having him chased away from the farm and then keeps him away through propaganda, having stories circulated about how he was a spy. He even blows up the windmill, blaming this sabatoge on Snowball. (Over time, the animals seem to even forget that the idea of the windmill was Snowball's in the first place!). He also uses the threat of 'Jones coming back' to coerce the animals into subjection.
The answer to your question stops here, but note that Napoleon doesn't rely simply on the animals' decisions to stay in power. He subtly seeks allies and alliances from the outside, such as the trading arrangements made with the neighbouring farms and a "middle man" to deliver whiskey. (Note that Orwell was implying England and Germany by neighbouring farmers Pilkington and Frederick.)
Finally, his subversive change in laws with no control by "checks and balances" is apparent throughout the story. Laws are written and rewritten to suit the fancy of the moment, and such changes escape all sanction whatsoever. In such a way the Seven Commandments are completely distorted from what they meant in the beginning. What was law becomes a simple game of semantics. Napoleon's ultimate sacrilege is when he has Squealer modify "All animals are equal" to "All animals are equal, but some are more equal than others."