The Torah states that Hashem told Moshe to bring his brother Aharon and his sons, from among Bnai Yisrael, to serve as Kohanim (שמות כ"ח:א'). The commentaries interpret this Posuk as signifying the transfer of the Kehunah,the priesthood, from Moshe to Aharon. As we know, during the first seven days of the installation of the Mishkan, Moshe served as the Kohein Gadol. Now, at Hashem's command, the priesthood was to be transferred to Aharon's family.
The Gemara in Zevachim (דף ק"ב.) states that Moshe lost the priesthood as a punishment for having refused previously to go on Hashem's mission to Paroh in Egypt, claiming that he had trouble speaking and could therefore not go and demand the release of the Jewish people. This angered Hashem who then said that Aharon the Leivi, Moshe's brother, who can speak well, will accompany him (שמות ד:י"ד-ט"ז). Rabbi Shimon Ben Yochai explains the identification of Aharon as a Leivi as implying that Hashem was telling Moshe that now Aharon is a Leivi, and he, Moshe, is a Kohein. But henceforth, Moshe will be a Leivi, and Aharon will be the Kohein. Many commentaries, however, hold that there was no transfer involved, and that Moshe was originally assigned to be the Kohein only for those seven days.
The Midrash Rabbah, (שמות פרשה ב') comments that Moshe's exclamation of "הנני", "Here I am", at the burning bush (שמות ג:ד') means "Here I am for the priesthood, here I am for the throne." Not only was Moshe interested in being the leader, he also wanted to be theKohein Gadol. But Hashem responded "אל תקרב הלום", "Do not come near" (שם פסוק ה'), suggesting that the priesthood is not yours, but is Aharon's. The Midrash also says that Moshe was upset when Hashem told him in our Posuk to bring Aharon forward; Hashem then told Moshe that He gave him His Torah. The Torah was in a way Moshe's compensation for losing the priesthood.
Moshe's jealousy of Aharon, and Hashem turning down Moshe for the priesthood, puzzles many commentaries.The Ibn Ezra, however, offers a straightforward answer to the problem. Moshe wished to serve Bnai Yisrael in whatever way possible, hence his desire to be the Kohein as well. However, Moshe bore the people's burdens. It was his primary duty to teach them Torah, which would leave him no time for sacrificial service. Moreover, Moshe's wife was Yisro's daughter, and his children thus lacked the pedigree for the job of priesthood. So it was not jealousy of Aharon, but a desire to serve the people that motivated Moshe, and Hashem did not reject him but felt he'd have no time for the job.
The Maggid of Dubno, (in his Ohel Yaakov) offers a psychological explanation of why Moshe was not chosen for the priesthood. A Kohein's task is to educate and direct the nation by personal example. Moshe, on his lofty level, stood beyond what the masses could attain. One who educates the nation by example must come from within the nation and must be part of it. Hashem did not give the Torah directly to the Jewish people but to Moshe, for at the time, the Jews were unfit to receive the Torah directly from Hashem. So Hashem made Moshe his agent. For this same reason, Hashem chose Aharon over Moshe because Moshe's spiritual level was so far above the people that he was unfit to direct them in terms of their normal lives. We can also interpret the expression in our Posuk instructing that Aharon be taken "from among Bnai Yisrael" (שמות כ"ח:א) in a similar fashion. Hashem commanded that a person be taken from among the nation, a Kohein who was part of the nation's body and soul. Such a person was Aharon, who could lead them on the path of righteousness.
The Abarbanel sees our Posuk as a command to announce "among the people" that Aharon had been chosen for the priesthood. This announcement, given by Moshe, would surely be accepted by the nation, because they would understand that it was Hashem's decree, for if given a choice, Moshe would have chosen one of his sons to be the Kohein. Finally, the Sifsei Kohen offers a totally different interpretation of this Posuk, noting that the Posuk refers to, the Kohein, Leivi, and the Yisrael. The Posuk thus shows us that by combining all the parts of the nation in the founding of the priesthood, Hashem invested the nation with added sanctity, and made it stronger.