Moses may not be the big kahuna in Leviticus, but a lot of what he says bolsters the authority of the Tabernacle priests. And the most important of these is the high priest, who just happens to be Moses's brother Aaron. It probably goes without saying that Leviticus does not have a law against nepotism.
So Aaron is the first high priest of Israel. He sets the standard for generations of priests to follow. But, um, he doesn't really say much. Instead, he shows his authority by what he does and what he wears—and in the eyes of God, he's fabulous. Stylin' ephod. The only Israelite allowed in the Holy of Holies. The guy who atones for the sins of the whole community on the annual Day of Atonement. More than anyone else in Israel, the high priest's got it goin' on.
Well, except for one teensy thing. Back in Exodus, Aaron sinned by indulging the people's desire to worship the golden calf. Oops. Leviticus makes it very clear that this is NOT OKAY by having God kill Aaron's oldest two sons in the middle of the sanctuary's opening ceremony. Watch and learn:
Now Aaron's sons, Nadab and Abihu, each took his censer, put fire in it, and laid incense on it; and they offered unholy fire before the Lord, such as he had not commanded them. And fire came out from the presence of the Lord and consumed them, and they died before the Lord. (10:1-2)
Ordinarily, burning the wrong kind of incense is not the sort of thing that would get a priest fired, let alone smoked. So what exactly did Nadab and Abihu do that is so bad?
The reference to unholy, literally "foreign," fire could be a more specific reference to bringing in fire from an area outside the Tabernacle that has not been set apart as holy to God. It's like wearing a baseball cap to graduation—sure, a cap covers the head as well as a mortarboard, but the principal will not approve.
The reference to foreignness also links the mistake to the corruption of the people by outsiders, which happens to be a theme of the story involving the half-Egyptian blasphemer, the son of Shelomith. And Leviticus might provide another clue when God offers the following friendly piece of advice after their deaths:
Drink no wine or strong drink, neither you nor your sons, when you enter the tent of meeting, that you may not die; it is a statute forever throughout your generations. (10:8)
God's command not to bring strong drink into the Tabernacle would seem to indicate that Nadab and Abihu offer the wrong incense because they're drunk. But verse 8 doesn't necessarily mean that they were looking on the wine when it was red. It might just be a means of making sure that priests keep their heads straight going forward. Banning alcohol from the sanctuary also distinguishes the Tabernacle from other Near Eastern temples, where incense and beer were a flowin'.
Whatever the exact reason, after God's holy fire burns Nadab and Abihu to death, Aaron and his other two remaining sons don't complete the sin offering ritual by eating the sacrificed meat. They know that dead bodies pollute the Tabernacle, even if that doofus Moses does not.