In Perek 61 of Vayikra, the Chumash discusses the Avoda performed in the Beit Hamikdash on Yom Kippur. It appears from Rashi's comments (to Pasuk 11 and Pasuk 71) that most of the Korbanot brought on Yom Kippur that are part of the special עבודת יום הכיפורים are מכפר only one sin namely, the sin of טומאת במקדש. All other sins of Bnai Yisrael are מכפר on the שעיר המשתלח, which is a goat which is thrown off a cliff and not even placed on the מזבח. Why is there so much complicated Avoda done to get rid of טומאה במקדש, while for all other Aveirot the שעיר המשתלח is sufficient.
The answer to this question may be that in truth a perfectly performed שעיר המשתלח could be מכפר all of Bnai Yisrael for all their sins (see Rambam Hilchot Teshuva Perek 1), however, in order to perform the ceremony of the שעיר המשתלח properly, the Beit Hamikdash in which the שעיר המשתלח is prepared, must be in perfect working order. By bringing Tuma into the Beit Hamikdash, Bnai Yisrael damage the Kedusha of the Mikdash. If the Kedusha of the Mikdash is flawed by Tuma, then the ceremony of the שעיר המשתלח cannot be performed to effect כפרה on behalf of Bnai Yisrael. For this reason, most of the Avoda done on Yom Kippur is performed to be מכפר only the sin of טומאה במקדש to ensure that the Beit Hamikdash be in perfect spiritual order before the actualשעיר המשתלח is brought and is hopefully מכפר all the other sins of Bnai Yisrael.
This fits in perfectly with the Pesukim in Perek 61. The first 91 Pesukim of the Perek discusses the Avoda of the פר חטאת, the גורל (lottery), the שעיר לה', and the Ketoret. Aharon says וידוי on the שעיר לה' and the פר חטאת, and he sprinkles the blood of the Par and שעיר on the פרוכת, and before the כפורת. Through this Avoda the Pasuk says he is (טז:טז) וכפר על הקודש מטומאת בני ישראל. Aharon's purpose is not to be מכפר the Jews for the sins of being מטמא the Kodesh, rather, to be מכפר the Kodesh from being Tamei, and he is repairing its Kedusha which it had lost. Then Aharon sprinkles the blood on the מזבח as the Pasuk says, (61:81) וכפר עליו...וטהרו וקדשו מטומאאת בני ישראל. Once again, Aharon is being מכפר the מזבח itself and not the Jew's sins. The next Pasuk says that with this Avoda (61:02) וכלה מכפר את המקדש ואת אהל מיעד ואת המזבח ותקריב את השעיר, "[Aharon] will finish cleaning the sanctuary and the מזבח and then he will bring the goat [לעזזאל]." This Pasuk tells us that up until this point, the goal of the Avoda was to cleanse the actual Mishkan of impurity. Only once this is done does the Pasuk tell us that Aharon is מקריב the שעיר המשלח which will be מכפר all the sins of Bnai Yisrael.
This idea can also explain a difficult Mishna in Shavuot. The Mishna says, ר' שמעון כשם שיחויו של שעיר המשתלח מכפר על ישראל כך וידוי של פר מכפר על הכהנים, "Rabi Shimon holds that just as the שעיר המשתלח atones for the Jews, so too the Par atones for the Kohanim." This raises a question, why do the Kohanim need to be מכפר with the Par? Why are they not simply included within the other Jews via the שעיר המתלח? It is easy to understand why the Kohanim would be separated out in regards to Tuma in the Mikdash since they are different than Bnai Yisrael in regards to their access to the Mikdash. However, why should the fact that they are Kohanim make them need to have a separate Korban to be מכפר for all their sins instead of being מכפר with the rest of Bnai Yisrael via the שעיר המשתלח. The answer, based on what we have already said, is that in order for the שעיר המשתלח to work properly, the Kohanim who perform the Avoda as well as the Beit Hamikdash itself must be spiritually perfect. For this reason, it is necessary to bring a separate Korban to be מכפר the Kohanim first, before bringing the שעיר המשתלח so that when the Kohanim bring the שעיר המשתלח it will effectively allow for forgiveness of Bnai Yisrael's sins.