Napoleon uses the dogs he has raised to chase Snowball, his rival, off the farm. At first, none of the other animals understand what is going on, but then they see the dogs following Napoleon and wagging their tails at him. They are "huge" and "fierce-looking" dogs in brass collars with studs.

After Snowball is gone, the fierce dogs surround Napoleon as he mounts the raised platform in the barn. He tells the other animals that the Sunday-morning meetings will come to an end. There will be no more debates about farm policy. From now on, all decisions will be made by a committee of pigs and announced to the other animals. In other words, Napoleon, with the backing of the dogs who are protecting him, has staged a coup and grabbed the power to run Animal Farm.

When some of the young pigs who are not in power start to protest, Napoleon uses the threat of force to hold onto his leadership. A quote that shows this is the following:

But suddenly the dogs sitting round Napoleon let out deep, menacing growls, and the pigs fell silent and sat down again.

Another quote that shows that Napoleon is using force—the dogs—to make sure his leadership is uncontested is as follows:

In these days Napoleon rarely appeared in public, but spent all his time in the farmhouse, which was guarded at each door by fierce-looking dogs. When he did emerge, it was in a ceremonial manner, with an escort of six dogs who closely surrounded him and growled if anyone came too near.

Later, when the hens protest their eggs being sold by refusing to lay, Napoleon

ordered the hens' rations to be stopped, and decreed that any animal giving so much as a grain of corn to a hen should be punished by death. The dogs saw to it that these orders were carried out.

Napoleon has set up a police state to intimidate anyone who challenges him. The police are the dogs. Nobody will dare speak up against Napoleon, because they fear the dogs will attack them.

Later, Napoleon stages a show trial where many animals are killed after making false confessions that they plotted with Snowball against the farm. We don't know who carries out the executions, but:

They [the animals who confess to crimes] were all slain on the spot. And so the tale of confessions and executions went on, until there was a pile of corpses lying before Napoleon's feet and the air was heavy with the smell of blood, which had been unknown there since the expulsion of Jones.

When it was all over, the remaining animals, except for the pigs and dogs, crept away in a body. They were shaken and miserable.

All of these quotes show how Napoleon uses force to impose his will on the other animals, who are too terrified to protest.