Can one learn Torah on Christmas day? I've heard there are a number of opinions on this: what are their sources? I've heard that Chasidim don't study anything Xmas night as well as the 6th of January. Is it just Christmas night or day as well?
The Nitei Gavriel which @yishai linked has an abundance of info of the customs and history of nittal nacht.
Rav Aviner brings a nice summary from his tshuvot (text):
"Question: Is it permissible to learn Torah on "Nittel Nacht" (Christmas Eve)?
Answer: There is a custom among some Chasidim not to learn Torah on "Nittel Nacht" in order not to contribute positive spiritual powers to idolatry. The Lubavitcher Rebbe related in a talk in the name of his father-in-law, the Rebbe Rayatz, that the 5th Lubavitcher Rebbe, the Rebbe Rashab, played chess on "Nittel Nacht" (Parashat Vayeshev, 1st Night of Chanukah 5750). And this was indeed the custom of the Lubavitcher Rebbe until midnight of "Nittel Nacht" (see Sha'arei Halachah U-Minhag vol 3 pp. 64-67. There is a famous picture of the Lubavitcher Rebbe and his father-in-law, the Friediker Rebbe, playing chess on Nittel Nacht). But our custom is that one needs to learn Torah on that night, just like every night. Some explain that the reason for not learning Torah of that night is that the Christians would become impassioned on their holiday, would go out and kill Jews (Ta'amei Ha-Minhagim p. 500). We therefore would need to stand guard instead of learning Torah (Ta'amei Ha-Minhagim p. 500). Baruch Hashem, Christians do not act this way today and we need to learn a lot of Torah.
By the way, it is told in the book "Siach Sarfei Kodesh" (p. 192) that when the last Lubavitcher Rebbe was sitting shiva, the Satmar Rebbe, Ha-Rav Yoel Teitelbaum, paid a shiva call and asked him: Why do we observe the night of "Nittel Nacht" according to the non-Jewish calendar (which is forbidden) instead of according to the Jewish calendar? The Lubavitcher Rebbe answered that we do not learn on that night in order not to give spiritual strength to the god of the Christians and we therefore observe it on the night that they celebrate, which is according to their calendar. And – he added – we must therefore be aware when they celebrate this holiday in the place we are located. In most places, their holiday is on December 25th, but it other places it is on a different day (For example, January 6th). In such places, that is the night of "Nittel Nacht".
But as we said, our custom is to learn a lot of Torah on that night. And Ha-Rav Moshe Sternbuch wrote that this custom was unknown in Lithuania and it is only a custom among Chasidim (Shut Teshuvot Ve-Hanhagot 1:551). The Chazon Ish would learn on "Nittel Nacht," and said that it was forbidden to waste time from learning on this night and he criticized those who did not learn on that night. The Steipler Gaon would also learn on "Nittel Nacht," but did so by heart so as not to upset those who have the custom not to learn. The Steipler Gaon also requested not to be informed when Nittel Nacht is so that he would not have to waste time from his learning (Orchot Rabbenu vol. 1 p. 193). And Ha-Rav Ovadiah Yosef has written that no such custom exists among Sefardic Jews (Shut Yabia Omer vol. 7 Yoreh Deah #20). The Lubavitcher Rebbe also writes that this custom does not exist in Eretz Yisrael (Sha'arei Halachah U-Minhag ibid. p. 67), even though it seems that some do have this custom in Eretz Israel."
The Netai Gavriel brings just about all there is to bring on the subject of Nitel. In 2:4 he brings two/three customs about the time of day or night it applies to. Although the custom is (perhaps more commonly) only from sunset or dusk until midnight, some (Chernobler, Belz, Bobov, Ger and Galitziyana) start from midday the day before (so that would be represented as the 24th midday until midnight - although the Stotshtein custom is to do that for both the 24th and the 6th).
In addition (2:1), there is a custom that varies the exact time of Nitel based on the season (Tekufas Teves), however there is no overlap between the two groups in those customs, so no one combines them.
So the upshot is any practice of Nitel by day is going to be from midday December 24th or January 6th at the earliest, and everyone stops by midnight (at least vis-a-vis learning Torah).
The answer is slightly different. It is a Jewish custom to learn Torah on the birthday or Yahrzeit of the deceased. So it became the Jewish custom not to learn Torah so as not to give the appearance of learning on behalf of Jesus on his birthday (which of course, Christmas is not--it was a pagan holiday long before Jesus that Christians adopted as a way of integrating Pagans into Chritianity).
Another strange custom that one sees among the very Orthodox is to write the word Christmas as Xmas. They dont realize that X was the Greek letter 'chi' which was written by Christians for "Christ" because of their regard for the holiness of the word. Just as we write H for the name of God.
there is an idea to specifically on the eve of the 25th to avoid learning then which only lasts until midnight. One source for this can be found in yom yom for the 17th of teves...
"The reason for not studying Torah on nittel-night, I heard from my father, is to avoid adding vitality.
My father once said: Those diligent students who begrudge those eight hours and cannot tear themselves away from study - I am not fond of them. This (abstaining from study) applies only until midnight." http://www.chabad.org/therebbe/article_cdo/aid/6037/jewish/Hayom-Yom-Tevet-17.htm
Here is also an online article that mentions this although I don't know what sicha it is referencing... https://crownheights.info/psa/49254/tonight-nittel-nacht/