See biography by J. D. McCallum (1939, repr. 1969).
He fulfilled a number of functions over the course of the Wilderness wanderings, from creating the plating to the altar out of the firepans of Korah's assembly to performing the ritual of the red heifer.
On Mount Hor he was clothed with the sacred vestments, which Moses took from off his father Aaron and put upon him as successor to his father in the high priest's office, which he held for more than twenty years. He took part with Moses in numbering the people, and assisted at the inauguration of Joshua.
He assisted in the distribution of the land after the conquest. When he died, he "was buried at Gibeah, which had been allotted to his son Phinehas in the hill country of Ephraim" (Joshua 24:33). The Hill of Phinehas related in the bible is associated with the location of the village of Awarta in the Samarian section of the current day West Bank. Due to the uncertain security situation, the Israel Defence Forces limits visits by Jews to one annual night close to the 5th of Shevat (around January-February) on the Hebrew calendar.
(Note: according to the Documentary Hypothesis, in the Elohist (E) document, "Eleazar" is the name of the son of Moses (see Exodus 18:4), who is however more commonly referred to with the name "Eliezer").