Berachos, Chapter 7, Mishnah 1. The seventh chapter of Maseches Berachos deals with the laws of Birkas HaMazon. The Mishnah opens with the case of three who ate together: they are obligated not only to recite a berachah, but also to make a zimun. The word 'zimun' literally means an invitation - those at the meal formally invite one another to bless.
Cases in which we do make a zimun even though there is a halachic problem involved:
"Achal demai" - Demai is produce purchased from an am ha'aretz, whom we do not trust to have separated ma'aser rishon from it. Terumah we do assume he separated, since amei ha'aretz were careful about terumah, but not necessarily about ma'aser; therefore one must separate ma'aser and ma'aser sheni from it, and there is an orderly procedure for doing so. The novelty of the Mishnah is that even one who ate demai before it was set right - we make a zimun on him. The reason is that "most amei ha'aretz do tithe," and only a minority fail to separate. For that reason it is forbidden to eat demai in the first place, but if one did eat it, since the majority do separate, the obligation of a berachah applies to him and one may include him in a zimun.
"U'ma'aser rishon shenitlah terumaso" - Ma'aser rishon is the first tenth that is separated and given to the Levi, and the Levi is obligated to separate from it terumas ma'aser - ten percent of the ma'aser, a hundredth of the total crop - and to give it to a kohen. If so, what is the novelty here? The case is one in which terumas ma'aser was indeed separated, but terumah gedolah (the roughly two percent given to the kohen from the crop) was not separated, and seemingly this should render the produce tevel, which is forbidden to eat. However, the case is one in which the Levi took the ma'aser from the crop before its processing was completed - before 'meiruach hakri,' the smoothing over of the pile of grain. Once the ma'aser rishon was separated before that stage, after the fact there is no obligation to separate terumah gedolah from it, and as long as terumas ma'aser was separated from it, it is permitted to be eaten.
"U'ma'aser sheni v'hekdesh shenifdu" - Ma'aser sheni must be eaten in Yerushalayim or redeemed, so that its money is brought up to Yerushalayim and food is bought and eaten there; hekdesh is something consecrated to the Beis HaMikdash for the upkeep of the building, which was afterwards redeemed. If they were redeemed, what problem is there in eating them? The novelty is that he did not add the fifth. One who redeems ma'aser sheni or hekdesh is obligated to redeem them for the principal plus an added fifth, meaning twenty five percent of the principal: produce worth one hundred is redeemed for one hundred and twenty five. And even though he did not add the fifth, the fifth does not hold up the redemption: the redemption is valid, the fifth remains a debt he owes, and in the meantime the ma'aser sheni and the hekdesh are considered redeemed and are permitted to be eaten.
"V'hashamash she'achal k'zayis" - The attendant who serves those seated at the meal, even though he has no place at the table, if he ate a k'zayis of bread, he joins the zimun.
"V'hakusi" - The Kusim were a group of people brought to Eretz Yisrael by the king of Ashur who converted, and there is a doubt whether their conversion was done properly. According to the Mishnah, the Kusi is considered a valid convert, and therefore "mezamnin aleihen" - on him and on all those listed here. However, after the period of the Mishnah the Kusim were considered non-Jews, and that is how we relate to them.
Cases in which we do not make a zimun:
From here the Mishnah moves on to cases in which one is not permitted to eat, and if he did eat them, we do not make a zimun on him:
"Achal tevel" - Tevel is any produce from which terumos and ma'asros were not properly separated. The Mishnah teaches that even tevel whose obligation in terumos and ma'asros is only rabbinic is forbidden to eat, and we do not make a zimun on it.
"U'ma'aser rishon shelo nitlah terumaso" - This does not mean that terumas ma'aser was not separated from it, those ten percent of the ma'aser given to the kohen, for it is obvious that such produce is forbidden to eat. Rather, the case is one in which terumas ma'aser was indeed separated, but terumah gedolah was not. In the first part of the Mishnah we learned that the failure to separate terumah gedolah does not hold things up, but there the case was one in which the ma'aser was taken before the processing of the grain was completed, before the meiruach was done. Here, after the processing was completed and the meiruach hakri was done, the grain is obligated in the separation of terumah gedolah. So when he separated ma'aser rishon at this stage and gave terumas ma'aser to the kohen, but did not separate terumah gedolah, those two percent, that is the problem, and the produce is forbidden to eat.
"U'ma'aser sheni v'hekdesh shelo nifdu" - This does not mean that they were not redeemed at all, for then it is obvious that they are forbidden to eat and that we neither make a zimun nor a berachah on them. The Gemara explains that ma'aser sheni must be redeemed with a coin that bears a stamp, a form and a marking; and if he redeemed it with an 'asimon,' a piece of metal that bears no stamp at all, this is not a fit means of redemption and the redemption is not valid. As for hekdesh, it does not require a coin bearing a form, and it can even be redeemed with a piece of metal - but the Mishnah is dealing with one who transferred the sanctity onto land, and land is not effective for the redemption of hekdesh. Under these circumstances the ma'aser sheni and the hekdesh are forbidden to eat, since their redemption was ineffective.
"V'hashamash she'achal pachos mik'zayis" - An attendant serving at the meal who ate less than a k'zayis does not join the zimun and cannot be part of it.
"V'hanochri" - This does not refer to an actual non-Jew, for it is obvious that he does not join. Rather, it refers to a convert in the midst of his conversion process: he has already been circumcised, "v'lo taval" - but he has not yet immersed in a mikvah. And as long as he has not immersed, his status is that of a non-Jew and he does not join the zimun.
And regarding all of these the Mishnah concludes: "Ein mezamnin aleihen".
In summary: This Mishnah deals with three who ate together, who are obligated in Birkas HaMazon and in zimun, and it lists two parallel sets of cases. Foods and participants on which we do make a zimun even though a halachic problem is involved: demai (since most amei ha'aretz do tithe), ma'aser rishon whose terumah was taken before the meiruach, ma'aser sheni and hekdesh that were redeemed without adding the fifth, an attendant who ate a k'zayis, and the Kusi. And in contrast, those on which we do not make a zimun: tevel, ma'aser rishon from which terumah gedolah was not separated after the meiruach, ma'aser sheni and hekdesh whose redemption was ineffective, an attendant who ate less than a k'zayis, and a convert who was circumcised but did not immerse.