CHATZITZOS UNDER THE FINGERNAILS [Netilas Yadayim: Chatzitzah]
Gemara
35a (Beraisa): The following are a Chatzitzah (separation) if they are dry, but not if they are wet - blood, ink, honey, and milk.
Chulin 106b (Beraisa): Any Chatzitzah that disqualifies Tevilah (immersion) also disqualifies Netilas Yadayim (washing the hands) for Chulin or washing the hands and feet in the Mikdash.
Eruvin 4b (R. Yitzchak): Chachamim decreed to disqualify a (Chatzitzah on the) majority when he is not Makpid (adamant), due to when he is Makpid, and a minority he is Makpid about due to a majority he is Makpid about.
Rishonim
Rif, Rambam and Rosh (Berachos 41b, Hilchos Berachos 6:4, Chulin 8:15): Any Chatzitzah that disqualifies Tevilah also disqualifies Netilas Yadayim.
Rambam (Hilchos Mikva'os 1:12): If there was on his skin or the Keli a Chatzitzah such as dough, pitch or similar things, even a drop like a mustard seed, if he is Makpid about it, the Tevilah was invalid. If he is not Makpid about it, the Tevilah was valid, unless the Chatzitzah was on the majority.
Rosh (Chulin 8:15): The following disqualify Netilas Yadayim: filth under the fingernail not opposite the skin, or dough under the fingernail even if it is opposite the skin, a bandage on his skin, Greek mud and potters' mud.
Hagahos Ashri (Berachos 2:11): Or Zaru'a says that one must be careful about scab in a wound or a boil that healed and there is a scab over it.
Tosfos (106b DH v'Chol): A Tosefta teaches that what is a Chatzitzah for Tevilah is not a Chatzitzah for Chulin! We can say that our Gemara discusses washing for eating. Chachamim were stringent about this like Tevilah. The Tosefta discusses touching Chulin, and teaches about even Chulin made Al Taharas Terumah. Chachamim were not so stringent about it. For eating, one must be careful about everything that is a Chatzitzah for Tevilah, e.g. dough under the fingernail. A small amount that he is not Makpid about is not a Chatzitzah, just like for Tevilah.
Sefer Chasidim (58): One should not grow long fingernails, lest mud enter underneath. Mud and dough are a Chazakah even not opposite the skin. Also, perhaps Chelev or Isurim will get caught underneath (and end up in his food), and he will transgress.
Poskim
Shulchan Aruch (OC 161:1): One must beware of a Chatzitzah in Netilas Yadayim. Any Chatzitzah for Tevilah is a Chatzitzah for Netilas Yadayim, e.g. filth under the fingernail not opposite the skin, dough under the fingernail even if it is opposite the skin, a bandage on his skin, Greek mud and potters' mud. We are not concerned for a Chatzitzah on the minority that he is not Makpid about:
Beis Yosef (DH v'Chasav): What is considered not opposite the skin? The Rashba says that it is if the fingernail was big and extended past the skin, and there was filth under that place past the skin. The Tur says that if he is not Makpid about, it is not a Chatzitzah. Surely he does not refer the last matter mentioned, i.e. filth opposite the skin. Rather, he refers to dough even not opposite the skin and filth not opposite the skin, which he said are a Chatzitzah. He teaches that this is only if he is Makpid about them. If not, since they are the minority of the hand, they are not a Chatzitzah.
Beis Yosef (DH Semak): Hagahos Maimoniyos says that it is proper to clean under the fingernails at the time of Netilas Yadayim. Some are lenient, for they considered this a minority that he is not Makpid about. Orchos Chayim says so. Semag cites a Tosefta (Mikva'os 6:10) that filth under the fingernail not opposite the skin, or mud or dough under the fingernail even if it is opposite the skin, are a Chatzitzah. R. Tam explains that we discuss mud like dough, that it is very stuck, e.g. potters' mud. Mishnayos in Mikva'os (9:3-4) say that filth under the fingernail, and even mud or dough are not a Chatzitzah. This is if there is so little that he is not Makpid about it. The Rashba says that some say that filth under the fingernail is a Chatzitzah for Tevilah and Netilas Yadayim. They bring a proof from the Tosefta. However, Tosfos says that they err. It is not a Chatzitzah for either, like the Mishnah says. This is because dough is very sticky. However, one always removes filth, for it decays and it does not stick; it cannot be a Chatzitzah. The Tosefta discusses potters' or builders' mud, which sticks like dough. Filth, which does not stick, is not a Chatzitzah. In Toras ha'Bayis, the Rashba says that Chatzitzos are things that stick very much to man and block water from coming (to his skin), e.g. Greek mud, potters' mud and mud with egg white mixed in, a scab, dry sap or dough under the fingernail are other things that stick greatly. He did not mention filth under the fingernail. However, in laws of Tevilah he said that filth under the fingernail not opposite the skin is a Chatzitzah. Perhaps we are lenient about Netilas Yadayim, which is mid'Rabanan, but not for Tevilah mid'Oraisa. In Toras ha'Bayis ha'Aruch he cites a Beraisa that filth under the fingernail not opposite the skin is a Chatzitzah, but in Hilchos Netilas Yadayim he says that it is not a Chatzitzah, for one is not Makpid about it and it does not stick so much, like the Mishnah says. We must say that this is when it is opposite the skin, but not opposite the skin it is a Chatzitzah, like for Tevilah.
Magen Avraham (4): Below (162:9), we say that something on the hands is a problem (for the water on the Chatzitzah is Tamei, and it will return to the hands and be Metamei them)! We must say that here we discuss something stuck to the hand. It is Batel to the hand. Alternatively, we discuss one who pours a Revi'is each time (so the water remains Tahor).
Mishnah Berurah (1): If anything sticks to the body, water cannot get there, so it invalidates Tevilah. Mid'Oraisa, this is for a majority he is Makpid about. Chachamim decreed about a Chatzitzah on the minority only if people are Makpid that it remain like that, even if at the time of Tevilah they are not Makpid. Something people are not normally Makpid is a Chatzitzah only if it covers the majority of the body. Even though Netilas Yadayim is mid'Rabanan, Chachamim enacted like Torah law. A Chatzitzah on the majority of the place that must be washed is a Chatzitzah even if people are not Makpid about it. One must remove from his hands any Chatzitzah that people are normally Makpid about, even if it is a minority of the hand. If after washing one found a Chatzitzah, even on the majority, we attribute that it came afterwards. l'Chatchilah, one should check if there is a Chatzitzah before washing.
Mishnah Berurah (2): The Shulchan Aruch discusses Tzo'eh, i.e. filth that accumulates under the fingernail. Presumably, real Tzo'eh (excrement), even if it was found opposite the skin, disqualifies Netilas Yadayim.
Kaf ha'Chayim (3): This is because excrement is Metamei the hands.
Kaf ha'Chayim (4): Women cut their nails before immersing because they cannot distinguish what is considered opposite the skin. Some say that the nail itself is a Chatzitzah if it extends past the skin The Shach (YD 198:25) says so if it is long and one intends to cut it. One should let it grow past the skin only due to Ones, e.g. a Shochet needs a long nail to check the knife, or a Mohel needs it to pull back the skin. They must check it carefully before washing.
Mishnah Berurah (6): Greek mud is pit mud. It is like potters' mud. It sticks very much, and water does not get underneath even if it is wet. The same applies to other kinds of dry mud. They are a Chatzitzah on the hand and under the fingernail, even opposite the skin. However a tiny amount that people are not Makpid about it is not a Chatzitzah anywhere. If one works with mud, e.g. he makes bricks or pots, surely it is not a Chatzitzah anywhere, unless most of his hand is covered. Since this is his profession, he is not Makpid about it.
Mishnah Berurah (7): The Tur holds that if one is not Makpid about something, it is not a Chatzitzah even if most people are Makpid. The Magen Avraham and other Acharonim say that regarding what the Shulchan Aruch mentioned, we follow the majority of people, who are Makpid.
Rema: This is why the custom is not to dig out mud under the fingernails for Netilas Yadayim, for it is like a minority that he is not Makpid about, for we are not Makpid about this for Netilah. If he was Makpid, he must dig it out. This is primary. Perhaps we are lenient about Chatzitzah for Netilah because some say that Chatzitzah does not apply to it. The first opinion is primary.
Taz (YD 198:14): Regarding Netilas Yadayim, we rely on R. Tam, who says that only potter's mud and similar things (that stick greatly) are a Chatzitzah.
Magen Avraham (5): Even though mud is a Chatzitzah even opposite the skin, people are not Makpid about this (so it is not a Chatzitzah). Most people are Makpid if it is not opposite the skin, even for filth, so it is a Chazakah even for one who is not Makpid.
Mishnah Berurah (9): Some are lenient about filth and Stam mud.
Magen Avraham (6): The Rema connotes that is a Chatzitzah when he is Makpid about it, even if most people are not. Pnei Yitzchak says that the Rashba was unsure about this, so we are lenient.
Mishnah Berurah (10): If he is Makpid about it but most people are not, we are stringent. It seems that this is only for potters' mud or dry mud. One need not be stringent about wet mud or earth under the fingernail opposite the skin, which does not stick so much, especially if he is Makpid about it only for Netilas Yadayim, but not during the rest of the day.
Kaf ha'Chayim (14): If the minority of a fingernail separated, one must cut it, for it is a Chatzitzah. If it is on Shabbos, he cannot cut it, so he does not bless Al Netilas Yadayim.