Contributed by: R. Ezra Mizrahi
The beginning of our parasha denotes the completion of the building of the Mishkan and the ceremonies that transpired afterwards.
As we know, the building of the Mishkan was a direct misva given to be "mitkaper", or atone, for the sin of the golden calf, "Het HaEgel".
On the onset of our parasha, we read of Aharon who is commanded to perform the following sacrifices in the newly built Mishkan:
1. Egel Ben Baqar - Le'Hatat (young bull for a sin-offering)
2. Ayil - Le'Olah (ram for an elevation-offering)
1. Se'ir Ezim - Le'Hatat (he-goat for a sin-offering)
2. Egel & Keves - Le'Olah (calf and sheep for an elevation-offering)
3. Shor & Ayil - Le'shlamim (bull and ram for a peace-offering)
One may quickly read through this part of the parasha, but if one carefully reads what sacrifices are being brought from whom, one can learn a great deal.
The three types of sacrifices are:
A. Le'Hatat: sin-offering, brought when one commits a sin
B. Le'Olah: typically brought to atone for bad thoughts that led to bad deeds.
C. Le'shlamim: typically brought as a donation
We know that the Mishkan was built to atone for the sin of the egel, and understandable, Aharon, who assisted in the creation of the golden egel, had to being an "egel" Le'Hatat, to atone for his actions.
Interestingly, Bnei Yisrael brought the "egel" for Olah, and not for Ha'tat. As you can see above, the he-goat was brought on behalf of Bnei Yisrael for Ha'tat.
Why the difference?
The Keli Yaqar answers beautifully. Aharon's sin was not in his thoughts but only in his actions. Of course Aharon did not truly believe in the egel in any way. However, he did sin in assisting in the creation of the egel. Therefore, he must bring an egel Le'Hatat to atone for his actions.
Bnei Yisrael sinned in their belief, and truly believed in the egel as a form of god, or liaison to Hashem. Therefore, they must bring their "egel" sacrifice as an Olah, which is typically brought to atone for misdeeds with bad thoughts.
Bnei Yisrael physically sinned by sacrificing animals (calves and sheep) to the egel, which is why their Hatat sacrifice (the sacrifice meant to atone for physical misdeeds) is an egel and keves, a calf and a sheep.