In what way did the Nesi'us of Elazar (in his capacity as 'N'si Nesi'ei ha'Levi') manifest itself?
Rashi and Targm Yonasan: All the appointments among the Levi'im were made by him 1 (through consultation with the Urim ve'Tumim - Targum Yonasan).
Yerushalmi, Shabbos, 10:3: #1: It means that he was the leader of the Nesi'ei ha'Levi. 2
Yerushalmi Shabbos, 10:3: #2: It means that he was the 'Amarkol' - he was in charge of all matters concerning the Levi'im - but he did have a number of tasks that he himself had to perform. 3
Oznayim la'Torah (citing Midrash Rabah): It means that he was the 'Amarkol' over the B'nei K'has only. 4
Rashi: As implied by the continuation of the Pasuk. Refer also to 4:27:1:2. (It is not clear however, seeing as Elazar was overall in charge of K'has, and Isamar, of Gershon and Merari - See 4:28 & 33 - what role Elyasaf ben La'el, Elitzafan ben Uziel and Tzuriel ben Avichayil, in their capacity as the Nasi of Gershon, K'has and Merari, respectively, played)? Refer also to 4:27:1:2 2
and to 4:16:2:1 and note 3.
And the Torah mentions him by the B'nei K'has because a. he was himself a Ben K'has and b. like the B'nei K'has, he carried holy articles. See Naso, 4:16.
See Naso, 4:16. See also Torah Temimah, note 11.
See Oznayim la'Torah, DH ''u'Nesi Nesi'ei ha'Levi' #1. See Oznayim la'Torah, who explains why, according to this explanation, the Torah dubs him "N'si Nesi'ei ha'Levi".