Eduyos, Chapter 1, Mishnah 8. This mishnah too opens an entirely new subject, unrelated to the one before it: the proper way to handle Terumah of karshinin (a type of legume known as vetch). The reason it appears here, like the other mishnayos in this chapter, is that Shammai disagrees with Hillel.
The halachic background:
Terumah is the portion of the produce that a person separates for the kohen, the first separation, and the kohen receives only about two percent of the crop. The verse says: "And I, behold, I have given you the charge of My terumos" - the owner is charged with a duty to safeguard the Terumah. This safeguarding includes two things: that the Terumah not be lost or wasted (as we find in the laws dependent on the Land, such as during the Shemittah year), and that it not become tamei.
Tumah is the main concern of our mishnah. Terumah produce is especially vulnerable to tumah, since it can become a "third", tamei to the third degree (a full explanation is too broad for this framework and is not our concern here). The important point for us is that the Sages decreed rabbinically that ordinary hands - hands that have not been washed - are considered "seconds", that is, a source of tumah of the second degree. What this means is that anyone coming to touch Terumah must first wash his hands, even if he is completely tahor and even if he has just come up from the mikvah. A remnant of this halachah is our washing of the hands before eating bread, which fundamentally originates from here.
One more introduction: the obligation of Terumah applies only to produce grown for human consumption. Produce grown for other purposes - flax for linen, or fodder for animals - has no status of Terumah, and there is no need to keep it in taharah.
What are karshinin:
Karshinin are a sort of poor man's pea. In principle they are edible, but people do not normally eat them, whereas in years of famine people do in fact eat them. Since the way of the world is to grow them as animal feed - in which case there is no obligation of Terumah - while at times they are eaten by people - in which case the obligation to separate does apply - the Sages imposed several stringencies upon them. The question of the mishnah, then, is how far those restrictions extend.
The views of the Tannaim in the mishnah:
Beis Shammai: "Karshinei terumah... shorin v'shafin b'taharah, u'ma'achilin b'tumah" - both soaking them in water, which is meant to soften them and make them fit to eat (and perhaps a person will eat them, even though in all likelihood the soaking is being done for the animals), and "shafin" - rubbing. The Bartenura explains this as rubbing them on the body in order to peel them, to scrape off and remove the dead skin, as was the practice in those days; and others explain that it is rubbing the seeds themselves, to remove the husks and the dirt from them. Both of these acts must be done in taharah, that is, with washing of the hands first. But one who feeds the produce to his animal need not wash his hands. The reason: at the time of soaking and rubbing, onlookers do not know whether the produce is destined to be eaten by a person, and therefore it must be kept in taharah; whereas at the time of feeding the animal it is evident to all that it is intended for animal feed, and taharah is no longer needed.
Beis Hillel: "Shorin b'taharah, v'shafin u'ma'achilin b'tumah" - only the soaking requires washing of the hands. The reason lies in the principle of hechsher: for tumah to take hold of a food, the food must first become wet, before the source of tumah touches it (this is the simple version of the matter; in truth it is somewhat more complex). Beis Hillel therefore hold: if a person wets the Terumah by soaking it, and in that very same act renders it tamei with unwashed hands - that is too much. In a single act he both makes it susceptible to tumah and touches it with unwashed hands, and this may not be done. But mere dry rubbing, as well as feeding the animal, are permitted in tumah and pose no problem.
Shammai: "Yei'achlu tzarid" - Shammai is stricter still: even at the time of feeding the animals, beyond what was said above, one must make sure the feeding is done while the produce is dry, so that it should not appear as though his unwashed hands are rendering it tamei right now.
Rabbi Akiva: "Kol ma'aseha b'tumah" - this is the most lenient view of all. In his opinion there is no room for any of these stringencies: the karshinin are intended for animals, and there is no reason whatsoever to impose restrictions on them. Anything from which Terumah is not separated for human consumption need not be dealt with at all, and even the soaking may be done without washing the hands first.
In summary: The halachah follows Beis Hillel, and therefore the Rambam rules that a person may do with his karshinin as he wishes even though they are Terumah, without washing his hands first - except for soaking. One who soaks them and wets them must wash his hands beforehand, for this is the threshold the Sages did not permit: to perform at one and the same time both the hechsher that makes it susceptible to tumah and the transmission of the tumah itself.