Gemara

(R. Chiya bar Ashi): For Mayim Rishonim (washing before the meal), one must tilt his hands upwards.

Support (Beraisa): For Mayim Rishonim, one must tilt his hands upwards, lest the water leave the palm and (the second pouring will not be Metaher the first water, and the first water will) return and be Metamei the hand.

Mishnah (Yadayim 2:3): Hands become Tamei and Tahor until the Perek (joint, e.g. wrist). If one poured the first water until the Perek, and the second time outside the Perek, and the water returned to his hand, his hand is Tahor. If he poured both of them outside the Perek, his hand is Tamei.

Rishonim

The Rif (Berachos 42b) brings our Gemara.

R. Yonah (DH she'Lo): Rav teaches that one must not (Bach - one must) lift his hands, lest the latter water become Tamei, and return to be Metamei his hands.

Bach (5): He is concerned only for the latter water. He holds that the latter water is Metaher all the first water, until and past the Perek. This is like Rashi's text, which is Metamei only when he was not particular to put latter water everywhere that first water was poured, i.e. also outside the Perek.

Mahari Ben Chaviv (cited in Beis Yosef OC 162 DH u'Mahari): Rav enacted to fix the problem (of Tamei water). R. Yonah enacted to lower the hands the entire time to totally evade the problem.

Rebuttal (Beis Yosef): This is wrong. R. Yonah explains Rav. He does not give his own enactment! Rather, R. Yonah was bothered why Rav did not say to lower the hands the entire time. If one need wash only the fingers, we could say that he teaches that one may raise them the entire time. He will be able to dry water that goes to the palm. However, the Rif obligates washing the entire palm. Some water will go past his wrist, under his sleeve, and he will be unable to dry it! Therefore, he explains that one (who wants to lift his hands) must lift them only slightly, so that water will not go past the wrist.

Defense (Taz OC 162:1): Rav agrees that it is best to lower the hands the entire time. He teaches that one who lifts them must lift them the entire time.

Rambam (Hilchos Berachos 6:16): For Mayim Rishonim, one must lift the hands up, lest water outside the Perek be Metamei the hands again.

The Rosh (Chulin 8:18) brings our Gemara and the Mishnah in Yadayim.

Rosh: One pours on his hands twice. The first water is Tamei, and the second is Metaher the first. Their Tum'ah and Taharah is until the Perek. Therefore, when he pours both outside the Perek, the second water is not Metaher the first there. If he lowered his hands and water outside the Perek returned to his hands, it is Metamei his hands. Therefore, one must lift the ends of his fingers up.

Beis Yosef (DH u'Mah she'Chosav Rabeinu d'Ta'ama...): The Rambam's and Rosh's text of the Mishnah says that hands are Metamei liquids only until the Perek. Water that touched outside the Perek is Tahor. Rashi's text says that if the first water was poured outside the Perek and the latter until the Perek, and the first water returned to the hand, it is Tamei. It seems that if also the latter water was outside the Perek, his hands would be Tahor, for he latter water would be Metaher the first. R. Shimshon cited Rashi to say so explicitly, but disagrees. He is Metamei the hands in every case, unless the first water was until the Perek and the latter was outside.

Hagahos Ashri (Chulin 8:43): We rinse meat a third time, to be Metaher the second water that touched the first. The Riva says that similarly, one who washes his hands must lift them, but one who washes three times need not.

Poskim

Shulchan Aruch (162:2): One who washes must lift his hands (Rema: i.e. the ends of the fingers) up, lest water outside the Perek be Metamei the hands again.

Question (Beis Yosef DH v'Yesh): According to the opinion that we are concerned only for water that became Tamei from the hand, left, and returns, why should one lift his hands from the beginning? It would be better to have them down the entire time, so the water will not become Tamei!

Answer #1 (Mahari Avuhav, cited in Beis Yosef): The Perek is not the wrist. Rather, it is where the fingers meet the palm. Some say that one need not wash there for Chulin, but water becomes Tamei there. If one would lower the hands, water would become Tamei there and be Metamei the fingers. This does not answer for the Rif (Berachos 41b), who requires washing there.

Answer #2 (Beis Yosef): Indeed, it helps to lower them from the beginning. Chachamim taught that if one raises them, he must keep them raised until drying them, lest water return to his hands. Why didn't other Meforshim say so? Why did Chachamim say 'one must lift them'? Perhaps there is an Asmachta from "Va'Ynatlem va'Ynas'em."

Beis Yosef (DH v'a'Ha): Rashi says that one must lift the ends of the fingers. It seems that he holds that only the fingers must be washed, therefore he did not say that the entire hand must be lifted.

Magen Avraham (1): If the Rema held like Rashi, who requires washing only the fingers, he would have said so in 161:4! Rather, he instructs not to raise the entire hand, lest water go past the wrist (see b:1:iv above).

Mishnah Berurah (2): If one did not lift them from the beginning, he need not wash again unless it seems that water left the Perek and returned.

Mishnah Berurah (3): One can lift his hands properly only when another pours on them. One who washes by himself should pour a Revi'is each time, for then he need not lift his hands (for the water remains Tahor). If he lacks water for this, he should lower his hands the entire time.

Rema: Likewise, it is fine if he lowers the hands from the beginning until the end. He must be careful not to raise the ends of the fingers at the beginning and later lower them, for the water outside the Perek will return and be Metamei the hands.

Mishnah Berurah (7,8): 'Until the end' means until the latter water. Afterwards he may raise them, even before drying them. However, one who raises his hands from the beginning may not lower them until drying them.

Mishnah Berurah (9): Some Acharonim oppose lowering the hands, lest water not get to the ends of the fingers. People are not careful to lift their hands, for we wash with a Revi'is each time.

Shulchan Aruch (ibid.): This is when he does not wash the entire hand until the wrist. If he washes until there, he need not lift his hands.

Rema: Some disagree.

Mishnah Berurah (11): Most require lifting the hands even in this case. One may rely on the Shulchan Aruch in pressed circumstances.

Shulchan Aruch (ibid.): Similarly, if he poured on both hands a Revi'is at once, there is no Tamei water at all, so he need not raise his hands.

Kaf ha'Chayim (1,3): According to Kabalah, at the time of the Berachah one should raise his hands even with his head, or at least the face, even if he poured a Revi'is each time.

Rema: Some say that if he pours on his hands three times, he need not be careful about all this. Our custom is to be lenient.

Beis Yosef (DH Kosav ha'Agur): The Or Zaru'a (1:73) says so. I do not understand the reason for this.

Magen Avraham (2): When he washes three times, none of the first water remains. Why isn't the first water Metamei the latter waters? Perhaps waters on the hands do not Metamei each other. Alternatively, when he pours three times, surely the second or third water reached all the first water. This is like the Rashba, but it does not help according to the Rosh.

Mishnah Berurah (16): Many are skeptical about this. One should pour a Revi'is each time to evade the need to be careful to lift his hands.