When we refer to nature, we using either mean the world of animals or a general personality. In Animal Farm, the animals refer to nature as themselves and their kind.
Old Major asks his fellow animals about nature in his famous dream speech:
But is this simply part of the order of nature? Is it because this land of ours is so poor that it cannot afford a decent life to those who dwell upon it? No, comrades, a thousand times no! (link p. 2, emphasis added)
When he refers to the order of nature, he is talking about the hierarchy of power both amongst animals and between animals and people. Old Major is expressing a frustration with the status quo, and the animals’ willingness to continue it. He suggests that it is not required that animals be subservient to (or less then, or slaves to) humans.
The humans, specifically Frederick and Pilkington, also use the supposed “laws of nature” to suggest the way animals ought to behave.
When time passed and the animals had evidently not starved to death, Frederick and Pilkington changed their tune and began to talk of the terrible wickedness that now nourished on Animal Farm. .. This was what came of rebelling against the laws of Nature, Frederick and Pilkington said. (p. 15)
Frederick and Pilkington spread rumors about cannibalism and torture on Animal Farm to reinforce the idea that something unnatural is happening there. It is against the laws of nature for animals to behave as humans do, according to their self-serving beliefs.
You can read the full text where these quotes were taken from here: http://msxnet.org/orwell/print/animal_farm.pdf