Gemara

(Beraisa): In Nisan of Yetzi'as Mitzrayim, Benei Yisrael left on the 15th, which was on Thursday.

(Beraisa): "Va'Yhi ba'Chodesh ha'Rishon. be'Echad la'Chodesh Hukam ha'Mishkan" - there were 10 special things about that day:

It was the first day that the Nesi'im offered Korbanos [for Chanukas ha'Mishkan]...

Ta'anis 17b (Beraisa): We do not [fast or] eulogize from Rosh Chodesh Nisan until the eighth, for then the Tamid was established (the Tzedukim were defeated). We do not [fast or] eulogize from the eighth until Pesach, for then Shavu'os was established (the Tzedukim conceded).

Rishonim

Tosfos (87b DH v'Oso): Since the 15th was Pesach, they took the animals [to prepare for the Korban] on Shabbos, which was the 10th of the month. It is called Shabbos ha'Gadol, due to the great miracle done on that day. The Medrash says that when they took their Pesachim that day, the Bechoros of the nations gathered and asked 'what are you doing?!' They answered 'it is Korban Pesach to Hash-m, who will kill the Bechoros of Mitzrayim.' The Bechoros went to their fathers and Paro, to request that they send away Yisrael. They did not consent. The Bechoros made a war, and killed many [of their fathers]. This is the meaning of "l'Makeh Mitzrayim bi'Vchoreihem."

Shibolei ha'Leket (205, citing Rashi's Sefer ha'Pardes): The Shabbos before Pesach is called Shabbos ha'Gadol, because that day Yisrael were commanded to take animals for Korban Pesach. They were afraid "if we will slaughter the deity of Mitzriyim, they will stone us!" Hash-m said "see the miracle that I will do." Hash-m commanded to take the animal from the 10th. The lambs were screaming, and the Mitzriyim heard, and their innards melted. They agonized, but they did not harm Yisrael at all, for they all became very sick due to anger. We call also the Shabbosos before the three festivals Shabbos ha'Gadol. Another reason is that people stay [in the Beis ha'Keneses] to hear the Drashah until after midday, close to Minchah, for they expound Halachos of Bi'ur, guarding the Matzos, Melachah on Yom Tov and on Chol ha'Mo'ed, the Seder, and telling the greatness of Hash-m who took us out of Mitzrayim.

Kol Bo (47): When the Mitzriyim saw Yisrael taking the animals for Korbanos, they took their swords and wanted to kill all the Yisre'elim. Hash-m sent great sicknesses on them, so they could not harm Yisrael.

Tur (OC 430): The Shabbos before Pesach is called Shabbos ha'Gadol, due to the great miracle done on that day. Yisrael were commanded to take animals for Korban Pesach on the 10th -- "be Asor la'Chodesh va'Yikchu Lachem Seh la'Bayis." They left Mitzrayim on Thursday, so the 10th was Shabbos. Everyone tied his Pesach to the leg of the bed. The Mitzriyim asked 'why are you doing this?!' They answered 'to slaughter it for Korban Pesach, like Hash-m commanded us." This set the teeth of the Mitzriyim on edge, that Yisrael will slaughter their god, but the Mitzriyim could not say anything.

Poskim

Shulchan Aruch (OC 429:2): We do not do Nefilas Apayim (say Tachanun) the entire month of Nisan.

Tur (OC 429): I found that from Rosh Chodesh Nisan until Isru Chag (the day after Pesach ends) we do not do Nefilas Apayim, fast or eulogize. Maseches Sofrim explains that we do not fast because on Rosh Chodesh Nisan the Mishkan was erected, and 12 Nesi'im offered Korbanos on the first 12 days, and each made a Yom Tov on his day. Therefore, we do not fast until Nisan passes. Firstborns fast on Erev Pesach.

Beis Yosef (DH v'Hachi): Maseches Sofrim (21:1-3) says that also, in the future the Beis ha'Mikdash will be built in Nisan, therefore we do not say Tachanun or fast. It seems that since the Mishkan was erected on Rosh Chodesh Nisan, and the next 12 days the Nesi'im offered, and then is Pesach, the majority passed in Kedushah, therefore we do not fast in the entire month.

Rebuttal (Bach DH Mipnei): What is the Beis Yosef's source to say that Maseches Sofrim argues with the Bavli, which holds that Nachson (the first Nasi) offered on Rosh Chodesh? Really, Maseches Sofrim agrees. Do not say that it mentioned Hakamas ha'Mishkan to forbid fasting on Rosh Chodesh. In any case we do not fast because Nachson offered that day, and it is Rosh Chodesh! Rather, it is to teach that the Nesi'im offered from the first through the 12th.

Birkei Yosef (3): The Beis Yosef is difficult. Mekom Shmuel answers that the 13th was a Yom Tov due to Shevet Levi, for then Aharon lit the Menorah. The Beis Yosef does not connote like this at all.

Kaf ha'Chayim (22): It is a good custom to say on each day (at the beginning of Nisan) the verse of the Nasi who offered that day, for it arouses the Kedushah of the day. On Nisan 13 we read from "b'Ha'aloscha" until "Ken Asah Es ha'Menorah."

Magen Avraham (3): The Nesi'im offered 12 days, and afterwards is Erev Pesach, Pesach and Isru Chag, and the majority of the month was in Kedushah.

Machatzis ha'Shekel: This connotes like the Beis Yosef, that the last Nasi offered on the 13th, and [immediately] afterwards is Erev Pesach. However, it is better to say that he holds like the Bach and Pri Chodosh. The majority of the month was in Kedushah, even though the 13th interrupted in between.

Beis Yosef (ibid.): The Tur connotes that we learn from Maseches Sofrim that we have Nefilas Apayim, fasting and eulogies after Isru Chag. Maseches Sofrim forbids fasting the entire month! I say that it is because the Gemara connotes that there is no Isur to fast in Nisan. Ta'anis 17b lists these days in Megilas Ta'anis (days on which one may not fast). In Rosh Hashanah 19b we say that Megilas Ta'anis was Batel! This is unlike the custom not to fast in Nisan. Therefore, the Roke'ach [whom the Tur cites] supports the custom from Maseches Sofrim, even though our custom is unlike Maseches Sofrim regarding from Isru Chag until the end of Nisan.

Kaf ha'Chayim (21): Ma'aseh Roke'ach on the Mishnayos (Reish Pesachim) says that in the future, the Beis ha'Mikdash will be built on Yom Tov, but the inauguration will be the week after Pesach, since we do not mix one Simchah with another. Therefore, we do not say Tachanun until Nisan finishes.

Taz (3): The custom is not to fast on any day on which we do not say Tachanun.

Gra (DH Ein): This is only a custom. According to law of, everything is permitted, for Megilas Ta'anis is Batel.

Shulchan Aruch (430:1): The Shabbos before Pesach is called Shabbos ha'Gadol, due to the miracle done on that day.

Beis Yosef (DH Shabbos): The Kolbo and Shibolei ha'Leket describe the miracle of Shabbos ha'Gadol like the Tur. If so, the miracle was from the 10th until Erev Pesach. All of these should be called great days! I answer that the primary miracle was at the beginning, which most set the Mitzriyim's teeth on edge. Once the first day passed, since they were used to it, they were used to it.

Taz (1): Many ask that if so, we should attribute the greatness to the 10th of Nisan even if it falls on a weekday! I heard from R. Moshe Charif that also the Jordan River split on the 10th of Nisan. If the 10th were called Gadol, people would think that the greatness is due to that miracle. Therefore, they called it Shabbos ha'Gadol, for the miracle of the Jordan was not on Shabbos. The Bach praised this answer.

Magen Avraham (1): Some ask that if so, we should make the 10th a Yom Tov, even if it falls on a weekday, just like other Mo'adim [which depend only on the day of the month]! Perhaps it is because they fixed a fast on the 10th of Nisan, for this is the day that Miryam died.

Kaf ha'Chayim (1): On Shabbos, the Mitzriyim believed that we will slaughter it, just we cannot do so today because it is Shabbos.

Birkei Yosef (1): Shirei Keneses ha'Gedolah brings from Maharil (Minhagim p.9) that it is called Shabbos ha'Gadol, for on that day we expound at length.

Rema: The custom is to say the Hagadah at Minchah from the beginning of "Avadim Hayinu" until "Lechaper Al Kol Avonoseinu." We cease saying Borchi Nafshi.

Gra (DH veha'Minhag): The custom is to read the Hagadah then, for this is when the miracle began. However, this is wrong. The Mechilta says "I might have thought [that we should tell our children about Yetzi'as Mitzrayim] on Rosh Chodesh. "That day" teaches that it is [Pesach, and] not Rosh Chodesh. One might have thought that "that day" is during the day [Erev Pesach, when we slaughter Korban Pesach]! "Due to this" teaches that it is when Pesach, Matzah and Maror are in front of you [to eat, i.e. on the night of the 15th].

Mishnah Berurah (2): We read the Hagadah on the Shabbos before Pesach, even if it is Erev Pesach. However, in such a year we expound the previous Shabbos.