In this chapter, several rebellions occur against Moses and Aaron.
It is with some irony that a former rebel, Aaron, is now himself subject to a challenge. Back in Num 12, Aaron (with Miriam) challenged Moses's leadership. At the time, I said that Aaron himself already held a position of substantial authority. Now we see, among other things, the Levites seeking to challenge Aaron's rights as the priest.
The exact nature of the protest is not entirely clear, but what we can see in v. 3 is that the challengers (Korah and other Levites, plus Dathan and Abiram who are sons of Reuben) want to "level the playing field" so that all the people may offer sacrifices and function as priests. The reason why I say it's not clear is because at first, it is a mixed group of Korah, Dathan and Eliab together with some "leaders of the congregation", but when Moses starts to address them, he is primarily referring to "you sons of Levi" who appear to be seeking the priesthood. So from the Levites, they seem to be seeking to overthrow Aaron, who is the priest. However, it says that they also rose up against Moses, who is not a priest but who instituted the priesthood. So there appear to be a couple things going on here, but the main thing is senior leaders of Israel and the Levites challenging Aaron and Moses (who appointed Aaron).
I am not without sympathy for their criticism, that all of the people "are holy" and "the LORD is in their midst". Ex 19:6, in particular, is a verse I have cited over and over as evidence that the LORD wishes to make the entire nation holy priests, which is precisely the criticism of Korah. The problem though is Korah's intent, which is to attain power. If his interest were in seeking the LORD, then that would be one thing, but I think this chapter makes it pretty clear that Korah is simply interested in removing Aaron and Moses from their positions of authority.
I do believe the LORD wishes to establish all the people as a sort of priesthood, but in the short term only a small number are allowed into the tabernacle. It's difficult for me to explain why the LORD would do it like this, but it's pretty clear that the Korahites do not have the authority to challenge the LORD's policy.
Dathan and Abiram, while lumped together with Korah, seem more focused on how they were denied access to the promised land. They primarily blame Moses, who is responsible for them as their leader. This is bitterly ironic since it was the rebellion of the people (including these leaders) that resulted in their condemnation to the wilderness. As with the Korahites, Dathan and Abiram think Moses is abusing his position and "lord it over us" (v. 13), as the NASB says. And as with the Korahites, their biggest issue is jealousy.
In the end, the LORD reaffirms Moses and Aaron by killing their enemies, swallowing them whole into the earth. Continuing with the bitter irony, the people blame Moses and Aaron for killing those leaders. Implicitly this shows us that the leaders must have been popular, and their accusations against Moses and Aaron were also probably popular. The LORD sends another plague to punish the people for gathering against Moses, and then Aaron is given an opportunity to exercise his priestly role. I think this is clearly intended to help establish Aaron's legitimacy, by giving him an opportunity to "atone" for the people in a very public way. The normal priestly obligations usually keep Aaron inside the tabernacle, which is largely concealed from the rest of the people. It's possible this is one reason why the people disrespect Aaron, because they are unaware of how he ministers on their behalf. I think that's one factor, but the bigger factor is clearly jealousy and ambition for the power that Aaron holds as the chief priest.
Briefly, I also an intrigued by this chapter's usage of censers. Censers had been only briefly mentioned before, almost casually stipulated in e.g. Ex 25:38 and 27:3. The first (and basically only) ritual use of censers is described in Lev 16, part of the Day of Atonement rituals. The casual side references of Exodus suggest that the usage of censers is either unimportant or very well-known to the author and his presumptive readers. I would bet more on the latter than the former, because surprisingly this chapter shows that many leaders have censers and not just the ones constructed for the tabernacle. In the end, those people were slain for offering impermissible incense, just like Nadab and Abihu in Lev 10. So it's interesting that all these people have censers, but using them is unregulated, possibly resulting in death if they do so improperly. Why do they all have censers? I can only imagine that there is some sort of customary or ritual use of censers that would have been more widely understood to ancient Israelites, but the rules for this have not been given to us. Instead, we only see them use censers as a sign of their piety, which is not accepted by the LORD and they are killed for it.