When Hashem asks Avraham Avinu to leave his homeland, his birthplace, and his family, and to go to the land that would eventually become Eretz Yisrael, He informs him that He intends to bless anyone that will bless him, and that He will curse anyone who will curse him (בראשית י"ב:ג'). The Gemara in Berachos (דף נ"ה.) cites this Posuk as a hint to the idea that one who is given a הכרב לש סוכ, a cup over which one recites a particular Beracha, but then refuses to make the proper Beracha, is among those whose life-span is shortened. It is clear from the comment of Rashi (שם בד"ה ואברכה) that the cup referred to here is the cup over which one recites Birchas HaMazon; the Shulchan Aruch (אורח חיים סימן ר"א סעיף ג') rules in accordance with this Gemara (שם). Rashi (שם) explains that the person who is given this cup in order to lead the Birchas HaMazon will bless the host who gives him the cup; this is presumably a reference to the Gemara earlier in Berachos (דף מ"ו.) which rules that it is proper for a guest to lead the Birchas HaMazon because he will include therein a blessing for his host, the text of which is spelled out there and is codified with slight alterations in the Shulchan Aruch (שם סעיף א') where this ruling is accepted. The Rambam (פרק ב' מהל' ברכות הלכה ז') notes that a guest may add on to his blessing for his host whatever he deems appropriate.
The Mishnah Berurah (שם ס"ק י"ד) explains that the connection to the Posuk in our Parsha (שם) is that when the guest who leads the Birchas HaMazon blesses his host, he himself will be blessed because the host is a descendant of Avraham Avinu, and by blessing him, the guest becomes deserving of the blessing promised by Hashem in the Posuk (שם) for those who bless Avraham (or, apparently, his descendants). If, however, the guest refuses to accept this cup and lead the Birchas HaMazon, and consequently refuses to bless his host, he will not receive that blessing promised by Hashem, and his life will be shortened. In the Shaar HaTziyun (שם אות י"ג), the Mishnah Berurah quotes from the Pri Megadim (באשל אברהם שם ס"ק ה') that although this punishment applies specifically to a guest who refuses this הכרב לש סוכ, as noted by the Magen Avraham (שם ס"ק ה'), one should in general, even if not a guest for a meal, try to get hold of a הכרב לש סוכ, that is, to be the one to recite a Beracha over a cup which is associated with a special Mitzvah.
The Mishnah Berurah then adds, though (שם בשער הציון אות י"ד), that he is unsure about whether or not this whole idea that a guest should not refuse when offered a הכרב לש סוכ even applies anymore today, because today, the practice is for every guest to include a special blessing in the Birchas HaMazon on behalf of his host, even if he is not holding a הכרב לש סוכ and leading the Birchas HaMazon. He thus goes on (שם אות ט"ו) to question the implication of the aforementioned Magen Avraham (שם) that the requirement to bless one's host applies only when reciting the Birchas HaMazon over a הכרב לש סוכ. The Kaf HaChaim (שם אות כ"ו) also suggests that today, when every person, not just the one holding the הכרב לש סוכ, says Birchas HaMazon for himself, as the Shulchan Aruch earlier (שם סימן קפ"ג סעיף ז') says is preferable, there is no problem for one to refuse the הכרב לש סוכ because he can still bless his host in his Birchas HaMazon anyway. The Aruch HaShulchan (שם סימן ר"א סעיף ג') likewise rules that the requirement to bless one's host applies today to every guest, even those not leading the Birchas HaMazon, because everybody says his own Birchas HaMazon. He thus concludes (שם) that perhaps today there is no particular need to even offer the honor of leading the Birchas HaMazon to a guest, as the Shulchan Aruch (שם סעיף א') requires in order that the blessing for the host will be recited, because it will be recited now in any case.
Based on an earlier Gemara in Berachos (דף נ"ג:) which indicates that it is proper, as explained by Rashi (שם בד"ה חטוף), to try and be the one to whom the כוס של ברכה is given, the Shulchan Aruch (שם סעיף ד') rules that one should make every effort to be the one who receives the הכרב לש סוכ in order to lead the Birchas HaMazon. As explained by the Magen Avraham (שם ס"ק ו'), the fact that this is referring to Birchas HaMazon is based on a comment of Rashi in Nazir (דף ס"ו: בד"ה חטוף), and the reason is that one gets a more immediate reward for reciting a Beracha than for hearing someone else's Beracha and answering Amen. The Mishnah Berurah in the Shaar HaTziyun (שם אות ט"ז), however, suggests again that since today everybody says his own Birchas HaMazon anyway, there may be no speedier reward for the one who leads the Birchas HaMazon, but he leaves the matter in doubt.
It is clear from all of the above that Birchas HaMazon is supposed to be recited with a הכרב לש סוכ, a cup which becomes associated with this Mitzvah. This requirement is found in the Gemara in Pesachim (דף ק"ה:) which infers, as interpreted by Rashi (שם בד"ה וש"מ) and by the Rashbam (שם בד"ה ש"מ), that Birchas HaMazon requires a סוכ, and it is spelled out even more clearly in a later Gemara in Pesachim (דף קי"ז:,ועיין שם בפירוש הרשב"ם בד"ה ש"מ). Another Gemara in Pesachim (דף ק"ז.) indicates, as explained by Rashi (שם בד"ה כוס) and by the Rashbam (שם בד"ה כוס), that the beverage to be used in this כוס for Birchas HaMazon must be wine; the Shulchan Aruch (או"ח סימן קפ"ב סעיף ב') rules accordingly. The Mishnah Berurah (שם ס"ק א') explains that the reason for requiring a כוס for Birchas HaMazon is that the proper way to honor and praise Hashem is to recite Berachos to Him over a כוס; the Kaf HaChaim (שם אות ב') also mentions this, and he cites Kabbalistic reasons for this as well.
According to Tosafos in Pesachim (דף ק"ה: בד"ה שמע מינה ברכה), it appears that this requirement applies even to an individual saying Birchas HaMazon on his own, and several sources and opinions are quoted there to document this, although, as Tosafos (שם) observes, the Minhag in the world is not to use a כוס for Birchas HaMazon unless three people are saying it together (as a מזומן). The implication, however, of the Rosh in Pesachim (פרק י' סימן י"ד) as well as, even more clearly, in Berachos (פרק ח' סימן ב') is that a כוס is in fact required even for an individual who recites Birchas HaMazon on his own. The Tur (או"ח שם) indeed rules this way, adding that one should not sit down to a meal unless he knows that he has, or can get, a cup of wine for Birchas HaMazon, except if wine is simply unobtainable; the Magen Avraham (שם ס"ק א'), however, questions this additional statement of the Tur (שם), and concludes that it is not necessary to go quite so far, and the Mishnah Berurah (שם ס"ק ב') asserts that the Vilna Gaon (ביאור הגר"א לסעיף א' שם בד"ה ולא) appears to agree. The Beis Yosef, commenting on the Tur (שם בד"ה והא), cites others who require even an individual saying Birchas HaMazon alone to use a כוס; the Bach (שם בד"ה ואחר) cites this as well in the name of the Maharshal (שו"ת מהרש"ל סימן כ"ג).
At the other extreme is the position of the Rambam (פרק ז' מהל' ברכות הלכה ט"ו) who holds that Birchas HaMazon actually requires no כוס at all; the Beis Yosef (שם בד"ה אבל) cites a source for this view, which, as he quotes from the Ran in Pesachim (דף כ"ו. בדפי הרי"ף בד"ה שמעת), is also the view of the Rif. The Beis Yosef (שם) also quotes from that Ran (שם) that the apparent requirement to have a כוס for Birchas HaMazon, as documented above, is intended to be understood as representing a מצוה מן המובחר, the best way to perform the Mitzvah, but it is not an absolute requirement. The Beis Yosef (שם) also quotes from the Rashba in Berachos (חדושי הרשב"א לדף נ"ב. בד"ה ולענין) that this is the position of the Rif and that the sources cited above which indicate that a כוס is needed for Birchas HaMazon are simply not accepted according to Halacha. The Rashba elsewhere (שו"ת הרשב"א חלק א' סימן שמ"ב) quotes that the Ramban as well held that it is certainly a מצוה מן המובחר to have a כוס for Birchas HaMazon, but it is not really required.
Many other Poskim, however, rule that although an individual saying Birchas HaMazon alone does not need a סוכ, three people saying it together, constituting a מזומן, do require a סוכ. The Hagahos Maimoniyos, commenting on the Rambam (שם אות ס'), quotes those who hold that way, as does the Beis Yosef (שם בד"ה וכתב ובד"ה ובמדרש), among others. The Shulchan Aruch (שם סעיף א') quotes all three of the above opinions, but while the Mishnah Berurah (שם ס"ק ד') states that the Shulchan Aruch (שם) offers no definitive decision, the Kaf HaChaim (שם אות א') asserts that the Shulchan Aruch (שם) really rules that no סוכ is needed at all for Birchas HaMazon. He then documents, however, that the Ari Zal, based on Kabbalistic sources, felt that if three people say Birchas HaMazon together, a סוכ is required. This is also the view of the Chayei Adam (כלל מ"ו סעיף ה'), the Kitzur Shulchan Aruch (סימן מ"ה סעיף א'), and others.
The Ramo (שם סעיף א') writes that it is a מצוה מן המובחר to recite Birchas HaMazon over a סוכ; the Kaf HaChaim (שם אות ג') suggests that this applies even to individuals, and the Aruch HaShulchan (שם סעיף ג') appears to concur, although he notes that in his part of the world, it is actually difficult to fulfill this because wine is not readily available and is very expensive. He also speaks (שם סעיף א') of a particular effort to use a סוכ for Birchas HaMazon that should be made on Shabbos and Yom Tov. The Shulchan Aruch HaRav (שם סעיף א') rules, however, that there is not even a מצוה מן המובחר for an individual to use a סוכ for Birchas HaMazon; the Mishnah Berurah (שם ס"ק ד') appears to agree. The Ramo (שם סעיף ב') also notes that if an individual does use a סוכ for his own Birchas HaMazon, he should, for Kabbalistic reasons, leave it on the table and not hold it, but the Mishnah Berurah (שם ס"ק ט"ז) concludes that today, no individual reciting Birchas HaMazon alone uses a סוכ at all. It may be added that the common practice currently is not to use a סוכ for Birchas HaMazon even if three people are saying it together; Rav Moshe Feinstein (שו"ת אגרות משה יורה דעה חלק ג' סימן נ"ב אות ג') discusses the reason for this.