In Chapter 9 of Animal Farm, how was the president of the New Republic elected?
In chapter 9, the farm is declared a Republic and a president is elected. Unlike traditional presidential races, where there are multiple candidates, the only candidate to choose from is Napoleon. Unsurprisingly, being the only candidate, Napoleon wins the election and becomes the new president.
What you'll notice from the text is that the details about the election are very scant. The narrator simply states that there was an election and that Napoleon won. No further details are given. This suggests that Napoleon's election is a mere formality in which the animals were not really given a choice about creating a new Republic or about who might govern it.
It becomes clear to the reader, then, that this election is a ruse by Napoleon to hide the fact that he is a dictator. He is using the illusion of choice to make the animals (and the reader) believe that life is better and fairer on the farm under him than it ever was under Jones.
We don't know the precise details of how the election occurred, and that speaks volumes about Manor Farm under Napoleon. Though formally a democratic republic, the farm has now become an Animalist dictatorship. Throughout history, many Communist dictatorships have called themselves democratic republics, or people's republics, and this is what Orwell is satirizing here. But whatever Napoleon and Squealer may say, we're left in no doubt that this was not an open contest, a free and fair election, or anything like one. The whole thing was just a gigantic sham designed to give Napoleon's dictatorship a veneer of democratic legitimacy. Whatever subsequently happens at Manor Farm, Napoleon can always say that the animals approved it when they chose to elect him as president. But he knows, and we know, that that is a total lie.
In the spring of that year Animal Farm was declared a Republic and so it was put forth that they needed to "elect" a president for it. All the animals voted for their new president, but the circumstances weren't exactly reminiscent of a democratically run government. There was only one candidate however- Napoleon so he won by an obvious unanimous landslide.