Uktzin, Chapter 3, Mishnah 10. Our mishnah picks up the thread that runs through the opening mishnayos of this chapter (mishnayos 1 through 3): the difference between foods that require hechsher, meaning that they must first come into contact with water before they are able to receive tumah, and foods that carry a tumah of their own from within, so that no water is needed to prepare them at all. Here the Mishnah applies that distinction to two surprising cases, and in the first one it teaches us a genuine chiddush.
The fat of a kosher animal that died
A neveilah is an animal of a permitted species whose life ended without valid shechitah, and such a carcass bears tumah by virtue of that status alone. Our instinct would be to say that whatever belongs to that body shares in the same tumah. The Mishnah, however, teaches otherwise: "chelev beheimah tehorah eino metamei tumas neveilos" - the chelev, that fat which the Torah places off limits for eating, does not transmit neveilah tumah.
Where do we see this? From the pasuk "vechelev neveilah vechelev treifah yeiaseh lechol melachah" - the fat of a neveilah and the fat of a treifah may be put to any use. The Torah is telling us that this fat remains usable, and usable even for purposes of kodesh. If the Torah grants it that kind of usability, it cannot be carrying the ordinary tumah of neveilah with it.
"Lefichach hu tzarich hechsher" - and precisely because no tumah is built into it, it takes its place among ordinary foods. Like any other food, it becomes susceptible to tumah only once it has first been wetted.
The fat of a non-kosher animal
And what of "chelev beheimah temei'ah", the fat of a non-kosher animal that died? Here the Mishnah rules the opposite: it does convey the tumah of a neveilah. There is no pasuk that carves it out and declares it usable, so it retains the very same status as the rest of the carcass, and it is tamei on its own account. "Lefichach eino tzarich hechsher" - therefore it requires no contact with water, because the tumah is already there within it.
Unclean fish and unclean locusts
The Mishnah continues with "dagim temei'im vechagavim temei'im", non-kosher fish and non-kosher locusts. (There are indeed species of locust that are kosher min haTorah, so the Mishnah is careful to speak specifically of the temei'im.) Regarding these, the ruling is "tzerichin machshavah bakefarim" - in the villages they require a specific intention before they can be reckoned as food at all.
The rule underlying this is that an item counts as ochel, and can therefore become tamei, only where people in that place actually eat it. Out in the villages no Jews eat such creatures, so they do not fall into the category of food on their own, and absent an act of thought they cannot receive tumah. In the sizable cities, with their large populations, there will be those who eat them, and there they qualify as food with no thought needed whatsoever. That is why the Mishnah frames its ruling so exactly: it is in the villages that these non-kosher fish and locusts call for a machshavah, a decision on the part of the owner that turns them into food.
So this mishnah leaves us with a neat pair of lessons. Tumah can come from within, as with the fat of a non-kosher carcass, in which case water adds nothing. And where tumah does not come from within, the item must first qualify as food, whether through hechsher of water or, for something the local population does not eat, through the intention of the one who owns it.