Korach's main complaint against Moshe was that there should not be a single leader for the nation, "For all the congregation are holy." (Bamidbar 16; 3). He was against what he saw as a dictatorial theocracy, and instead claimed to be advocating equality for all. This, however, was only a pretext for his true motivation, which was to become the new leader of the nation. This is why he accepted Moshe's challenge that he and all those who joined his rebellion should bring an incense offering, and let G-d choose who the leader should be. Despite his claims of equality, Korach personified the famous line from ‘Animal Farm’ that "all animals are equal, but some are more equal than others".
Not only Korach, but all those who followed him also openly espoused equality, but in reality were aiming for personal political power. This is why the Mishna (Ethics of the Fathers 5; 17) states: "What is an example of an argument that is not for the sake of heaven? The argument of Korach and his congregation." Were his followers supporting Korach, the Mishna should have said the argument of Korach and Moshe. We see from here that there was more fighting between Korach and his followers than between them and Moshe, and this is because they each wanted to be in charge.
Maor VaShemesh explains the Talmud (Nedarim 39b) based on this idea. "When Korach confronted Moshe, the sun and the moon went before G-d and said, 'Master of the Universe, if you do justice for the son of Amram (Moshe) we will continue to shine. But if not we will cease to shine'". Maor VaShemesh asks why the sun and moon were particularly involved in this argument? He bases his answer on the Talmud (Chullin 60b) which relates that originally G-d created the sun and the moon both the same size, as the verse states "G-d made the two great luminaries" (Bereishis 1; 16). However the moon complained to G-d that two kings cannot both rule equally. Therefore G-d told the moon to make itself smaller, as the verse continues, "the big light and the small light". We see from here that G-d agreed fundamentally with the moon's claim that there can only be one ruler. Therefore, when Korach tried to claim that everyone was equal, and there was no need for a single ruler, both the sun and moon objected.
The Torah contains a story which deals with the inherent risks of having two equal leaders. Cain and Abel were originally the only two sons of Adam and Eve. The Torah states: "And Cain said to Abel his brother" (ibid. 4; 8). The Midrash (Bereishis Rabba 22; 7) explains that Cain's pretext for killing Abel was making a pact with him. Cain was to take the entire earth as his inheritance, and Abel was to have all the chattel. Cain would claim that the ground on which Abel was standing belonged to him, and Abel claimed that Cain's clothes belonged to him. Abel told Cain to remove his clothes, and Cain told Abel to fly in the air. Eventually Cain resolved the argument by killing his brother.
Korach hadn't learnt the lesson that it is not possible for two leaders to divide their kingdom. He continued to uphold the argument of Cain that everyone should have equal rights to govern. Therefore Moshe's challenge to Korach was the same as that of Cain and Abel. Just as they brought offerings to G-d, so too Moshe told Korach and his congregation to each bring an offering, and in that manner let G-d decide who should be the rightful leader. Moshe's prayer was "Do not turn to their gift offering", which is clearly a reminder of the earlier verse "G-d turned to Abel and his offering, but to Cain and to his offering He did not turn".
Rav Yonasan Eibeschitz in his book Tiferes Yonasan takes this idea to the point where he writes that Korach's soul was actually a reincarnation of the soul of Cain, and Moshe was a reincarnation of Abel. At this point in history the record is set straight, that ultimately righteousness, and serving G-d with a pure heart and good intentions, will triumph over might and egotistical power lust.