Taanis, Chapter 2, Mishnah 6. This Mishnah addresses the question of whether the kohanim assigned to serve in the Beis HaMikdash during a week in which fast days for rain fall are obligated to fast, and in what manner.
The division of the kohanim into mishmaros and batei avos:
The kohanim are divided into twenty-four mishmaros, that is, twenty-four groups, each of which serves during a different week over the course of twenty-four weeks. Each day of the week is further divided among batei avos - seven different sub-groups, according to most opinions. It emerges that the beis av serving on a given day has one status, while the mishmar serving that entire week has another status, and this distinction is the foundation of our Mishnah.
The opinion of Rabbi Yehoshua:
"Shalosh ta'aniyos harishonos" - the first three fasts that the Sages decreed upon the community because rain was withheld, on which one fasts only during the daytime. On these, the members of the mishmar "mis'anin velo mashlimin" - they fast, but they do not complete the fast, and toward the end of the day they may eat. Even though it is not their day to serve during that week, at times the work in the Beis HaMikdash increases and assistance is needed from the other batei avos; since they may be called upon, they are not required to complete the fast. The members of the beis av, however, who actually serve on that day, "lo hayu mis'anin kelal" - they would not fast at all.
"Shalosh shenios" - the second set of three fasts: the members of the mishmar "mis'anin u'mashlimin" - they fast and complete the fast; and the members of the beis av of that day "mis'anin velo mashlimin" - they fast, but do not complete it.
"Sheva acharonos" - the final seven fasts, which are the most stringent: "eilu va'eilu mis'anin u'mashlimin" - both the members of the mishmar of that week and the members of the beis av of that day fast and complete the fast.
All of the above is "divrei Rabbi Yehoshua" - the words of Rabbi Yehoshua.
The opinion of the Sages:
The Sages are more lenient in each of these categories:
"Shalosh ta'aniyos harishonos" - "eilu va'eilu lo hayu mis'anin kelal" - both the members of the mishmar of that week and the members of the beis av would not fast at all.
"Shalosh shenios" - the members of the mishmar "mis'anin velo mashlimin" - they fast but do not complete the fast; while the members of the beis av, who serve in the Beis HaMikdash on that day, "lo hayu mis'anin kelal" - they would not fast at all.
"Sheva acharonos" - the members of the mishmar "mis'anin u'mashlimin" - they fast and complete the fast; and the members of the beis av, the kohanim serving on that very day, "mis'anin velo mashlimin" - they fast but do not complete it, so that they will be able to carry out their service in the Beis HaMikdash.
In summary: In this Mishnah we learned the obligation of the kohanim to fast on the fasts for rain, based on the distinction between the members of the mishmar of that week and the members of the beis av serving on that particular day. According to Rabbi Yehoshua, the stringency increases from the first three fasts until the final seven, on which both groups fast and complete the fast; the Sages, however, are more lenient at every stage, and even on the final seven the members of the beis av do not complete their fast, because of the needs of the service in the Beis HaMikdash.