ויקחו בני-אהרן נדב ואביהוא איש מחתתו ויתנו בהן אש וישימו עליה קטרת ויקרבו לפני יקוק אש זרה אשר לא צוה אתם: ותצא אש מלפני יקוק ותאכל אותם וימתו לפני יקוק
The sons of Aharon, Nadav and Avihu, each took their fire pan, put fire in it, placed on it incense, and offered before Hashem an alien fire that they were not commanded [to bring]. A fire then went forth from before Hashem and consumed them and they died before Hashem
Right after the inauguration of the Mishkan, the Tabernacle, tragedy strikes. The entire Jewish people are overjoyed that their hard work has paid off. They’ve finally built the Mishkan, and Hashem has shown that His Divine Presence is with them. Two of the sons of Aharon, wanting to express their gratitude, offered a voluntary incense offering. Their plan backfires and a fire comes forth and kills them. The verses seem to indicate that their sin was bringing an unwanted offering. However, Chazal indicate that there were other sins which caused their deaths.
Rav Dessler compiles fifteen different possibilities for what their sin was. Some examples are as follows: some say that they issued a ruling in front of their teacher, a major sign of disrespect. Some say that they entered the Holy of Holies, a place reserved only for the Kohel Gadol once a year on Yom Kippur. Some say it’s because they didn’t consult one another before offering this incense. Others say their sin was that they offered the incense while drunk. Another opinion is that they refused to get married, feeling that no woman was suitable for them. One opinion cites an incident which occurred, where Nadav and Avihu were walking behind Moshe and Aharon and were asking: “When are these old men going to die and we’ll be the leaders of the people”?  This seemingly disrespectful attitude led to their demise.
It’s very strange that the different opinions of their sin are so distant from each other. One doesn’t really have anything to do with the other. However, Rav Dessler teaches a fundamental principle when understanding the words of Chazal in the context of Aggadah, their non-legal teachings: all the different opinions are all teaching the same message, just from different perspectives. If we look deeply at what the different opinions are saying, they are all coming from the same root. The essence of the sin of Nadav and Avihu was according to their incredibly lofty level, they were missing the perfection of the attribute of humility.
This explains why they issued a ruling in front of their teacher, as well as why they didn’t consult one another. It explains why they relied on their level to enter the Holy of Holies and offer an incense that they weren’t commanded to bring. Even though they weren’t yet commanded against serving in the Temple while drunk, they should have had the proper awe of Heaven to realize it was inappropriate. The Chofetz Chaim explains similarly the opinion that they were asking when Moshe and Aharon would die. It’s not that they were looking forward to their deaths. Rather, they were concerned that it would happen eventually, and they would be called upon to lead the people. They were confirming to themselves whether they were worthy of this position. While their intentions were good, on their level it was presumptuous to think that they would be the next leaders. They should have been more humble.
The attribute of humility is very hard to acquire. Humility is often understood as denying our abilities. This is incorrect. Moshe wrote that he was the humblest person to ever live. According to this definition, writing that would be a contradiction to what’s being written. Rather, a proper understanding is being aware and acknowledging our abilities, but not taking credit for them. People are born with innate qualities, and it’s not through any fault of their own. Recognizing that everything comes from Hashem is true humility. We see even the sons of Aharon were lacking in this quality. However, it’s definitely something attainable. It just takes a lifetime of work.
 Based on Michtav MeEliyahu Volume 2 p. 244-250
 Leviticus 10:1-2
 ibid 9:24
 Yoma 53a; Eruvin 63a; Midrash Tanchuma 6:6; Toras Kohanim Shemini Mechilta D’Miluim § 32
 Vayikra Rabbah 20:8; Midrash Tanchuma loc. cit. The third sin he lists is what the verse says, they brought an offering that they weren’t commanded to bring (see Gur Aryeh to Leviticus loc. cit. who says that those that say sins one and six agree that the sin was bringing an alien fire, but their deaths were immediate because of the sins listed). The fourth he lists is they inappropriately brought the fire from elsewhere in the Temple
 ibid and Toras Kohanim loc. cit. This is inferred from the words איש מחתתו, each his fire pan, implying that they didn’t do it together
 Vayikra Rabbah 20:9; Midrash Tanchumah loc. cit. This is from the juxtaposition of their death to the command in verse 9 which prohibits Temple service while drunk. Sins seven, eight, and nine are that they were lacking one of the required garments for service (the me’il), they didn’t do the ritual hand and feet washing, and they didn’t have children (ibid)
 Some say out loud (and in Midrash Tanchuma loc. cit. it says they said this in front of Moshe and Aharon) while some say they merely thought this
 Sanhedrin 52a; Vayikra Rabbah 20:10; Midrash Tanchumah loc. cit. The twelfth sin is from Exodus 24:11, that they saw a vision of Hashem and carelessly ate and drank. The thirteenth is from Ramban to Leviticus 10:2 with the explanation of the Rikanti that they had intentions to the attribute of judgment and they should have had intentions to the attribute of compassion. The fourteenth is from the Rikanti quoting the Zohar that they served purely out of love, without any inclusion of awe. The fifteenth (ibid) is their fire was lacking in some spiritual way
 See Michtav MeEliyahu Volume 3, the letter titled מחלקת חז”ל
 In his commentary to Toras Kohanim
 See Michtav MeEliyahu Volume 2 p.249 for an explanation to how this fits the other sins listed
 It’s one of the last steps in Mesilas Yesharim’s guide to achieving prophecy, after being free of sin and having only pure intentions
 Numbers 12:3