There is a long-standing debate among the commentaries as to whether the Kohen Gadol performed every part of the Temple service on Yom Kippur, or if other kohanim participated in performing parts of the service that are not directly connected with the unique avodat Yom ha-Kippurim (Day of Atonement service). The Ramban argues that the second perek of Massekhet Yoma, which begins on our daf, appears to support the position that other kohanim were involved as well, since the entire discussion in the perek revolves around how to choose which kohen will perform what part of the avodah. Others argue that this is simply a discussion of the procedure that took place on other days, and it is brought here as a tangent, since the last Mishna in the first perek discussed terumat ha-deshen, or cleaning the ash off of the altar.
In any case, the Mishna on our daf teaches that there was a race every morning in the Temple, as all of the kohanim interested in performing the terumat ha-deshen would line up and race up the ramp to the top of the altar. The one who arrived first had the honor of cleaning the ash.
Initially, that was the procedure; however, an incident occurred where both of them were equal as they were running and ascending on the ramp, and one of them shoved another and he fell and his leg was broken. And once the court saw that people were coming to potential danger, they instituted that priests would remove ashes from the altar only by means of a lottery. There were four lotteries there, in the Temple, on a daily basis to determine the priests privileged to perform the various services, and this, determining which priest would remove the ashes, was the first lottery.
The Me’iri explains that this curious method of choosing the kohen, the race, stemmed from the fear that no one would want to perform this particular avodah, as cleaning the ash from the altar hardly seems to be a great honor. Nevertheless, other commentaries ask how such a contest could be instituted in the Temple, a place where an atmosphere of solemnity should prevail. The Tosafot Yeshanim explain that this wasn’t a normal race. In fact, the kohanim were obligated to walk up the ramp as they ordinarily did, placing heel in front of toe and again, heel in front of toe. The kohen who succeeded in doing this most quickly in a dignified manner was crowned the winner and rewarded with the opportunity to clear the altar to begin the day’s Temple service.