Regarding the death of Nadav and Avihu, the Torah says it happened when they brought a foreign / unauthorized fire into the Mishkan.
Concerning their death, Rashi brings up Rabbi Eliezer's opinion that they died because they gave halachic decisons in front of their teacher Moses. Rashi also brings Rabbi Ishmael who says they died because they entered the Sanctuary intoxicated by wine.
Obviously Nadav and Avihu were not a couple of drunkards who made a mistake after partying once the Mishkan was inagurated, for later on Rashi tells us how Moshe said that "Now I see that they (Nadav and Avihu) were greater than I or you!” regarding himself and Aaron.
So what really happened here? What is this "foreign fire" that they brought, and what is the significance of Rabbi Eliezer's and Rabbi Ishmael's comments regarding this situation?