Before us is the second mishnah in chapter 8 of Maseches Terumos, which is a direct continuation of the previous mishnah. It opens with the connective vav - "vechulam" - referring back to the cases listed there.
The Bartenura explains that "all of them" refers to the woman who thought she was still married to a kohen and was eating Terumah, and to the slave who ate Terumah because he thought he was still counted among the kohen's household - and then they were informed that the woman had been divorced, that the slave's master had died, and the like, so that they are no longer permitted to eat this Terumah.
The question the mishnah addresses: what is the status of the Terumah food already in their mouths at that moment - must they spit it out? On the one hand, when they put the food into their mouths it was a perfectly legitimate act: all the information available to them indicated that they were permitted to eat Terumah, and they did nothing improper. On the other hand, in the previous mishnah we learned that these very same people must pay the principal and the fifth in order to gain atonement for having eaten Terumah unintentionally, which makes Rabbi Eliezer's position somewhat difficult to reconcile.
The wording of the mishnah and the dispute of the Tannaim:
"Vechulam shehaysah terumah besoch pihem" - the Terumah food was inside their mouths and they were chewing it at the moment they were informed that they were no longer fit to eat this Terumah:
Rabbi Eliezer says: "yivla'u" - the process of eating has already begun, and they need not spit out the food in their mouths.
Rabbi Yehoshua says: "yifletu" - since these people are not permitted to eat Terumah, from the moment they learn that they are disqualified from eating it they must spit out the food, even though they began eating it thinking they were permitted to do so.
However, a kohen who thought he was a kohen but is in fact a chalal was never fit to eat this Terumah at all. In such a case he may not swallow the Terumah; he must spit it out - even according to Rabbi Eliezer.
The second clause: tumah that occurs in the middle of eating:
From here the mishnah moves from Terumah eaten unintentionally by someone forbidden to eat it, to a different matter: the prohibition of eating Terumah when the person eating is tamei or when the Terumah is tamei. Exactly the same principle operates here: the person began eating with full permission, and in the middle of chewing something happened that changed the situation. This is an even sharper and clearer case than the previous ones.
"Amru lo nitmeisa" - a kohen is in the middle of eating his Terumah food, and while he is chewing someone taps him on the shoulder and says: "I forgot, I am a zav, I am tamei - and now you have become tamei too." When he put the food into his mouth and began chewing it, everything was entirely in order, and only in the middle of eating did he become tamei. Rabbi Eliezer says: "yivla", and Rabbi Yehoshua says: "yiflot".
"Venitmeis terumah" - a similar case governed by the same laws, except that here it is not the kohen who became tamei, but the Terumah itself that became tamei while still in his mouth (the zav stuck his finger into the kohen's mouth, or tamei food entered his mouth, and the like). Rabbi Eliezer says "yivla", since at the moment he put the food into his mouth he was tahor and the food was tahor; and Rabbi Yehoshua says "yiflot".
The cases in which all agree that he must spit it out:
"Tamei hayisa" - they told him that already when he entered the house he had passed through a place where there was a corpse and had become tamei, so that at the moment he put the Terumah food into his mouth it was already disqualified, and it was never permitted to him to eat it.
"U'temei'ah hayesah Terumah" - the Terumah food had become tamei hours earlier, even before he put it into his mouth.
"O noda shehu tevel" - here we are not dealing specifically with Terumah and a kohen, but with any person: while eating, he found out that the food in his mouth is tevel, meaning that it was not properly tithed, and therefore it is not fit to be eaten.
"U'ma'aser rishon shelo nitlah terumaso" - ma'aser rishon is the gift given to the Levi, and it too requires tithing: the Levi gives a tenth of his ma'aser to the kohen, and that tenth of the tenth is called terumas ma'aser. Ma'aser rishon that has reached the Levi's hands and from which terumas ma'aser has not been taken is forbidden to eat until it is tithed.
Ma'aser sheni and hekdesh that were not redeemed - ma'aser sheni is food that in years 1, 2, 4 and 5 must be brought up to Jerusalem and eaten there, or whose sanctity must be redeemed onto a coin; and if he did not do so and it is outside Jerusalem, eating it is forbidden. Hekdesh is food that was dedicated to the Beis HaMikdash and became holy, and an ordinary person may not eat it until it is redeemed or given to the Beis HaMikdash for its use or for sale.
"Ta'am ta'am pishpesh" - a person who is eating ordinary food and tastes in his mouth the taste of a pishpesh. A pishpesh is apparently a bedbug, and in modern Hebrew it means a flea - and for the purposes of halachah there is no difference: there is an insect in his mouth, and the food in his mouth is not fit to be eaten.
What all these cases have in common is this: these foods were never fit and were never permitted to be eaten. For that reason, in all of these circumstances even Rabbi Eliezer agrees - the one who permitted swallowing food that he began to eat permissibly, such as Terumah that became tamei after he put it into his mouth - and according to all opinions he must spit the food out.
Practical halachah:
Here is a common case: a person ate meat and forgot, and made a berachah on a piece of chocolate, or even already put it into his mouth, and only then remembered. In all these cases he may not eat it, and one may not say "this is only a rabbinic prohibition," "time has already passed," and the like. Anything forbidden to eat, even rabbinically, may not be eaten. And if as a result this turns out to be a berachah levatalah - that is the law, and he must spit the food out.