What is Nittle Nacht? What does the name mean? What is its significance? Which Halachic authorities speak about it, and what do they say? What is the actual date for it? How is it effected this year when it comes out on Shabbos?
"Nittle nacht" is the Yiddish reference to the night going into Christmas. (It was observed on different dates, depending on if you lived in a Catholic/Protestant country, or an Eastern Orthodox [Christian] country; the former have Christmas on December 25th, thus "nittel nacht" starting at sunset December 24th; the latter have a different calendar.). There were customs in Eastern Europe about men not learning Torah and/or women not going to the mikva that night, at least not until after midnight.
The simple meaning of the term is "nacht" means night; "nittel" as in "neonatal"; "birth night", as it was the night observed by Christians for the birth of Jesus.
The simplest explanation is found in Rabbi Dr. Leiman's lecture on yutorah.org; that the night of Christmas often had a lot of non-Jews roaming the streets (occasionally drunk), and thus it was a dangerous time to be outside. Therefore, women were told not to go the mikva, and men not to go out to study halls or synagogues to study (most people didn't have a lot of study material at home). Past midnight, everyone was in church (or headed home), and it was safer outside.
In Hayom Yom (17 Teves), the reason given (in the name of R' Shalom Dovber Schneersohn, fifth Lubavitcher Rebbe) is "to avoid adding vitality." The point is that the person whose birthday they're celebrating on this day was a Jew, and since on a person's birthday his mazal (spiritual source) is stronger, we don't want the spiritual benefits generated by our Torah learning to be diverted towards strengthening his mazal.
I've heard different customs about what to do when it comes out on Friday night. Some say that Shabbos overrides this issue, since as the Zohar, recited by some on Friday night, says, וכל דינין מתעברין מנה - all harsh judgments go away in the face of Shabbos. Others keep this custom even on Friday night, though they will sing zemiros, tell stories of tzaddikim, or the like, instead of regular divrei Torah.
Nitel Nacht is the eve of the non-Jewish holiday celebrating the birth of the Nazarene (see Divrei Yatziv O.C.2 240:1). According to some Nitel is associated with the Latin for being born, nacht of course meaning "night". (Nitei Gavriel on Nittel 1:2, see also Halichos Chaim, Moadim u'Zmanim, nittel 1 note 2).
December 25th for the Eastern Church falls out on January 7th, so in eastern Europe the observance coincided with the tekufas Teves.
The practice was, and in some communities is, to refrain from learning Torah on the night of Nitel Nact. Some explain it as a pragmatic concern since anti-Jewish violence was a common feature of non-Jewish religious holidays. I believe the Chasam Sofer hypothesizes that it was to be rested in order to wake up and learn at midnight so the situation wouldn't occur thant non-Jews were up worshiping while the Jews slept. Another view is that we do not wish to strengthen the negative spiritual forces brought about by the celebration, a view that seems to be particularly prevalent among those who observe the practice.
Some observe the practices of Nittel Nacht even if it falls out on Friday night, others do not.
Nitei Gavriel has one of the more extensive discussions of the topic, including the various other practices associated with it and questions arising from the fact that their holiday is observed at different times. He suggests that many authorities did not mention the practice out of fear of the government.
There are a number of theories as to where the term "Nittel", or "Nittel Nacht" originate from. Some suggest that it comes from the Latin "Natale Dominus", meaning, "The birth of our God". Others surmise that the use of the word Nittel etymologically comes from the Hebrew "natal", meaning "to have been hanged". Nittel can also be construed as the Hebrew word for "being taken away". Alternatively, it may simply be a derogatory nickname that was used to refer to Jesus in all circumstances. The more likely meaning of the term Nittel is that of an acronym for "Nolad Yeshu Tet L'tevet", meaning, "Jesus was born on the ninth of Tevet."