Gemara

(Mishnah): An [army] encampment is exempt from washing.

(Abaye): This applies only to Mayim Rishonim (washing before eating), but Mayim Acharonim (after eating, before Birkas ha'Mazon) is a Chovah (obligation).

Question (R. Chiya bar Ashi): Why is Mayim Acharonim obligatory?

Answer: This is due to Sodom salt, which blinds the eyes.

Question (Rav Acha brei d'Rava): If one was Kayil (measured out salt), must he wash his hands?

Answer (Rav Ashi): Surely, he must!

Berachos 53b (Beraisa): "V'Hiskadishtem" alludes to Mayim Rishonim. "Vi'Hyisem Kedoshim" alludes to Mayim Acharonim. "Ki Kadosh" alludes to oil. "Ani Hash-m Elokeichim" alludes to the Berachah [Birkas ha'Mazon].

Chulin 105a (Rav Idi bar Avin): Mayim Rishonim is a Mitzvah. Mayim Acharonim is a Chovah.

Rishonim

Rambam (Hilchos Berachos 6:3): One may wash after Kol (any) bread with salt in it, lest there is Sodom salt in it, or a salt whose nature is like Sodom salt, and he will pass his hands over his eyes and get blinded. Therefore one must wash after every meal due to the salt. In an encampment one is exempt from washing beforehand, for they are busy with fighting, but he must wash afterwards, due to danger.

Rebuttal (Ra'avad): This is unlike the Gemara, which says that if one was Kayil (measured out) salt, he must wash. The Rambam ruined the wording.

Kesef Mishneh: If our text of the Rambam is correct, what is his source that one must wash after eating] any bread with salt in it? (Note: It seems that he questions the source that the bread must have salt in it.) Also, why did the Rambam omit washing after measuring salt? Also, what was the Ra'avad's objection? The Rambam's law is unlike Kayil! Perhaps the Ra'avad understood that the Rambam explains that Kayil refers to Achilah (eating). If we would alter the text of the Rambam to say 'Kol Es ha'Melach', just like it says in Eruvin and Chulin, why did the Ra'avad object? Perhaps he holds that the Rambam should have changed the wording to make this clear. This is not difficult, for the Rambam cites the Gemara verbatim.

Rosh (Berachos 8:6): Our Gemara connotes that Mayim Acharonim is due to Kedushah. Chulin 105b [which says like Eruvin 17b] connotes that it is due to danger of Sodom salt! We can say that it is a Mitzvah due to Kedushah, and a Chovah due to danger. Rav Hai Gaon says that it is a Mitzvah due to Kedushah only for the Mevarech (the one who blesses aloud to exempt the others), for he mentions Hash-m's name. The others wash only due to danger, but not for Kedushah. They need not wash before Birkas ha'Mazon. Nowadays the custom is not to wash Mayim Acharonim, for Sodom salt is not commonly found. Also the Mevarech does not wash. Since we do not follow the verse regarding oil, also we do not follow it for Mayim Acharonim. The hands are not considered filthy, since our custom is not to wash and we are not particular about this. We find that oil is essential only for those particular about it (Berachos 42a). Even though we learn it from a verse, it is a mere Asmachta. However, if one is an Istnis (finicky) and usually washes after the meal, for him his hands are [considered] filthy, and he must wash before Birkas ha'Mazon.

Tosfos (17b DH Mayim): Nowadays the custom is not to wash Mayim Acharonim, for Sodom salt is not common. Alternatively, we do not normally dip our fingers in salt after eating.

Tosfos (Chulin 105a DH Mayim): In Berachos, we expound Mayim Rishonim and Mayim Acharonim from one verse. It is an Asmachta. Mayim Acharonim is due to Sodom salt. We never find a Mitzvah or obligation of oil.

Poskim

Shulchan Aruch (181:1): Mayim Acharonim is a Chovah.

Beis Yosef (DH v'Hem): Rashi explains that Chachamim said to eat salt after every food. R. Yonah (Berachos 40b DH v'Iy) says that the Rambam says that even though nowadays we do not have Sodom salt, one should be concerned for other salt with the same nature. R. Yerucham wrote in the name of the Rif that even if one did not eat salt, he must wash because Mayim Acharonim killed a person (Yoma 83b).

Mishnah Berurah (1): Mayim Acharonim is because the hands are dirty from eating, and they are disqualified from blessing. Even if one does not bless, rather he hears Birkas ha'Mazon, he is obligated. Even one whose hands are not dirty must wash, due to concern for Sodom salt.

Mishnah Berurah (2): If one has limited water, he should use the minimal amount for Mayim Rishonim, in order to leave a little water for Mayim Acharonim. If he has only enough for Mayim Rishonim, that has precedence, for nowadays Sodom salt is not commonly found, so it is less obligatory than Mayim Rishonim. Some say that l'Chatchilah one should not eat unless he will have also Mayim Acharonim. In any case, it is not a greater Chiyuv than Mayim Rishonim, so one need not go more than four Mil in front [in the direction he was traveling] or backtrack one Mil to get water for it. If he will not find water within this Shi'ur, even so he may eat and bless Birkas ha'Mazon afterwards, and whenever he gets water, he will wash due to concern for Sodom salt.

Chachmas Shlomo: The Gemara calls Mayim Rishonim a Mitzvah, to teach that one may not choose not to fulfill it. It calls Mayim Acharonim a Chovah to teach that one must exert more for it than Mayim Rishonim, i.e. even more than four Mil [in front] or one Mil [in back].

Note: The Gemara did not discuss how far one must go for it. Perhaps Chachmas Shlomo holds that there is no limit.

Kaf ha'Chayim (1): All Divrei Chachamim are based on secrets, but they clothe them in Peshat (simple, revealed meaning). The Zohar says that after one eats, he must give a portion to the Sitra Acha (Satan), i.e. Mayim Acharonim, the filth. This expels the Sitra Acha. This is why Chachamim were so adamant about it. Also, one should not touch his moustache while his hands are wet from Mayim Acharonim.

Kaf ha'Chayim (5): Even if one did not eat bread, if his hands are dirty, he must wash before saying a Berachah Acharonah.

Kaf ha'Chayim (6): One should wash with only a small amount of Mayim Acharonim [for it goes to the Sitra Acha]. If one's hands, mouth or moustache are very dirty, he should first wash them with intent to eat more, eat more bread or other food, and afterwards wash Mayim Acharonim.

Kaf ha'Chayim (7): If water is not available, he may clean with earth or a pebble, like it says in OC 4:22.

Shulchan Aruch (10): Some have the custom not to wash Mayim Acharonim.

Magen Avraham (8): This is because nowadays Sodom salt is not commonly found, and these people are not concerned for dirty hands.

Gra: Tosfos, the Rosh and Tur say that if one is particular, [he must wash]. Then, "vi'Hyisem Kedoshim" applies. This is wrong. Tosfos in Eruvin is difficult. There we discuss an encampment, and we exempt them only from Mayim Rishonim, but not from Mayim Acharonim, for it is a Chovah! If the reason of Sodom salt does not apply, they are exempt in an encampment, but obligated elsewhere, just like Mayim Rishonim! The Rosh, R. Yerucham and Raya Mehemna say so. This is primary.

Mishnah Berurah (22): The Gra holds that also nowadays one must wash. Sodom salt and filth apply even nowadays. The Magen Avraham says so in the name of Mekubalim. The Maharshal was stringent and the Birkei Yosef is very stringent.

Kaf ha'Chayim (27): Yalkut Reuveni (Vayikra) says that Iyov's punishments came because he did not wash Mayim Acharonim. The Zohar and Ari Zal say that it is obligatory also nowadays.

Shulchan Aruch (ibid.): Even they hold that if one is an Istnis and usually washes after the meal, for him his hands are [considered] filthy, and he must wash before Birkas ha'Mazon.

Magen Avraham (9): Even if one wants to bless on wine and his hands are dirty, he must clean them first.

Mishnah Berurah (23): The Magen Avraham's law applies to an Istnis.

Magen Avraham (10): The Tur connotes that only the Mevarech must wash. The Mekubalim say that all must be careful about Mayim Acharonim. The Yam Shel Shlomo would wash the cup and his hands with water.

Mishnah Berurah (24): One may not interrupt between Mayim Acharonim and Birkas ha'Mazon even for Divrei Torah.