Gemara

18b (Beraisa #1): If one washed or immersed his hands, they are Tehorim only if he intended.

Contradiction (Beraisa #2): His hands are Tehorim whether or not he intended.

Answer (Rav Nachman): Ma'aser requires intent. Chulin does not.

Question: What is the source that Chulin does not require intent?

Answer #1 (Mishnah): If a wave containing 40 Sa'im of water separated from the sea and fell on a person and on Kelim, they are Tehorim.

Suggestion: Just like the Kelim had no intent for Taharah, also the person!

Rejection: No. The case is, he was waiting for such a wave to fall on him. The Kelim are like the man. Just like he must intend to become Tahor, he must also intend for them to become Tehorim.

Answer #2 (Mishnah): If Reuven's hands were Tamei and his produce fell into an irrigation channel, and he stuck his hands in and took it out, his hands become Tehorim and the produce is not Huchshar. If he wanted to wash his hands, his hands become Tehorim and the produce is Huchshar.

Question (Rava - Beraisa): If a man immersed with intention to eat Chulin in Taharah, he becomes Tahor for Chulin, but he may not eat Ma'aser.

This shows that intention is required even for Chulin!

Answer (Rav Nachman): No. The Beraisa teaches that even though he became Tahor regarding Chulin, he may not eat Ma'aser.

Question (Rava - Beraisa): If a man immersed without intention, it is as if he did not immerse.

Suggestion: It is as if he did not immerse at all.

Answer (Rav Nachman): No. It is as if he did not immerse regarding Ma'aser, but he became Tahor for Chulin.

Rava doubted the validity of these answers until he found a Beraisa supporting Rav Nachman.

(Beraisa #3): If a man immersed without intention, he is Tahor regarding Chulin, but he may not eat Ma'aser.

Chulin 31a (Rav): If a Nidah was immersed unwillingly, she is permitted to her husband, but forbidden to eat Terumah;

(R. Yochanan): She is forbidden even to her husband.

Question (Abaye): Beraisa #3 refutes R. Yochanan!

Answer (Rav Yosef): R. Yochanan holds like another Tana:

(Beraisa - R. Yonason ben Yosef): "V'Chubas Shenis" is extra, to equate the second Tevilah (of a garment with Tzara'as) to the first. Just like the first Tevilah requires intent ('V'Tzivah ha'Kohen v'Chivso"), also the second.

106b (Rav): One may wash his hands in the morning and stipulate that he will guard their Taharah the entire day (so he may eat without washing again).

Pesachim 115b (Rav Chisda): (Even though) one washed his hands for the first dipping (Karpas), he washes for the second dipping (Maror).

(Rabanan): This refers to other days, but here (Seder night) he knows that he will dip again, so he need not wash again.

Objection (Rav Papa): No, it refers to here. On other days, why should we assume that he dips twice?!

Counter-question: If it refers to here, why must he wash again? He already washed (knowing that he will dip again)!

Answer: Since we say Hagadah and Hallel in between, perhaps he was distracted and touched (something that is Metamei the hands).

Berachos 43a (Beraisa): Even though people already washed one hand (when they drank), they wash both hands to eat bread.

Rishonim

Rambam (Hilchos She'ar Avos ha'Tum'ah 13:2): One who immersed without intent is Tahor for Chulin but forbidden to eat Ma'aser Sheni unless he immerses with intent for Ma'aser. Similarly, if one washed his hands or immersed them without intent they are Tahor for Chulin but not for Ma'aser.

Ran (Chulin 7a DH v'Hachi): The Rambam rules like Rav against R. Yochanan. This is reasonable, for Rav Nachman holds like him. Also, R. Yochanan's opinion is not even brought in Chagigah.

Kesef Mishneh: The Rambam holds that the Stam Gemara, not Rav Nachman, brings the Mishnah of a wave to show that intent is not needed for Chulin.

Rambam (Hilchos Mikva'os 1:8): Everyone who immerses must intend for Tevilah. If he did not, the Tevilah helps (only) for Chulin.

Rambam (9:17): If a wave of 40 Se'ah separated from the sea and fell on a person or Kelim, they are Tehorim for Chulin, for Chulin does not require intent.

Hagahos Ashri (Chulin 2:8): Even though normally Chulin does not require intent, the Halachah follows R. Yochanan (against Rav), who requires intent to permit a Nidah to her husband). However, if Rachel forcibly immersed Leah, Rachel's intent suffices.

Tosfos (Pesachim 115a DH Kol): One who washed for drinking must wash both hands before eating, because the first washing was not for Kedushah.

Tosfos (115b DH Tzarich): The second dipping (Maror) is after ha'Motzi. When we discuss washing for the second dipping, it means washing for ha'Motzi, which is also for Maror.

Question (Tosfos 115b DH Asuchi): Obviously we must wash again. The first washing was not for Kedushah, rather, lest he Metamei liquid (on the Karpas).

Answer (Tosfos): Since we do not require intent for Chulin, washing for any reason would suffice for bread (if his mind were not diverted).

Question: This contradicts what Tosfos said above (DH Kol)!

Answer (Gra OC 158:7 DH Notal): The Gemara in Berachos requires intent for Chulin. The questioner in Pesachim does not require intent; the Gemara answered according to his premise. Tosfos himself rules like R. Yochanan against Rav. Also, the Halachah follows Rava against Rav Nachman.

Poskim

Shulchan Aruch (OC 158:7): If one washed his hands to eat a food dipped in a liquid and afterwards he wants to eat bread, some say that that the first washing does not count, and all the more so if he washed without intent to eat.

Beis Yosef (DH mi'Divrei): Tosfos (Pesachim DH Asuchi) connotes that he must wash again. This requires investigation.

Rebuttal (Darchei Moshe): Tosfos says so only regarding Seder night, for he interrupts between the first washing and the meal to say Hagadah and Hallel (but normally, he need not wash again). Also the Gemara explicitly says so!

Mishnah Berurah (28): The opinion that requires washing again holds that one need not wash for dipped foods nowadays (that we are all Temei'im).

Rema: If he did not divert his mind (from eating), he washes again without a Berachah.

Magen Avraham (13): Some say that intent is not needed (159:13), hence he need not wash again. Therefore, he washes without a Berachah.

Mishnah Berurah (30): If he ceased to guard his hands, all agree that he washes again with a Berachah.

Shulchan Aruch (159:13): L'Chatchilah, (when washing for bread) one should intend to permit eating. L'Chatchilah, it suffices for the Nosen (the one who pours the water) to intend, even if the Notel (whose hands are washed) does not intend.

Beis Yosef (DH Aval Kol and DH v'Da): A Mishnah (Yadayim 1:5) allows a Cheresh, lunatic or child to pour the water. In the Tosefta (1:7), Chachamim say that intent of the Nosen or the Notel suffices; R. Yosi says that his hands are Tamei unless both had intent. R. Shimshon says that R. Yosi agrees to the Mishnah; a Cheresh, lunatic or child can have intent. The Rashba says that the Halachah follows Chachamim. Alternatively, R. Yosi discusses Terumah, but Chulin does not need intent. In Chagigah we do not require intent of the Nosen or the Notel. We must say that the Tosefta discusses when one of them had intent to teach the extremity of R. Yosi's opinion, but Chachamim do not require intent at all. All the Poskim hold like this.

Rashba (brought in Gra (ibid.) and Beis Yosef DH Aval ha'Rashba): The Halachah follows Chachamim, who require intent of the Nosen or the Notel. All require intent to eat Chulin. Even Rav himself (Chulin 106b) requires one who washes in the morning for the entire day to stipulate, even though he guards his hands. The Beraisa about produce that fell in water teaches that no intent for Tevilah is needed to touch Chulin (Al Taharas Terumah, or regular Chulin, which could become Tamei through hands that are a Rishon l'Tum'ah). The enactment to wash before eating bread was for the sake of Terumah, which requires intent. If Chachamim did not require intent at all, the Tosefta should have taught this, for it is better to teach the extremity of the lenient opinion! The Beraisa that says that his hands are Tehorim whether or not he intended, i.e. the Notel, but the Nosen intended.

Rebuttal (Gra ibid., citing the Ro'oh): The Tosefta discusses Terumah. Rav requires stipulating, for otherwise he will not guard his hands well. This is correct; washing is not more stringent than Tevilah, which does not require intent. The Gemara (Chagigah 18b) says that no one requires washing to touch Chulin. It does not establish the Mishnah to discuss hands that are Rishonim.

Beis Yosef (DH ul'Inyan): Even though the Poskim do not require any intent for Chulin, not for the Nosen or the Notel, since they did not explicitly say that intent is not needed, we follow the Rashba.

Mishnah Berurah (75): Most say that if one washed without intent he need not wash again. Surely one may rely on them if he has no more water.

Kaf ha'Chayim (78 and 64): If one washed without intent and did not yet dry his hands, he washes again and blesses. If he already dried them, he should touch a covered part of the body, then wash again and bless. The same applies to one who ceased to guard his hands.

Kaf ha'Chayim (79): Intent is not needed for dipped foods.

Mishnah Berurah (76): Intent of the Nosen does not suffice if he is less than six years old.