Shulchan Aruch (429:2) states, "We don’t say tachanun the entire month of Nisan." The Mishnah Berurah explains, "The twelve nesi'im brought korbanos [on the first twelve days of Nisan], and on the day that one brings a korban, it is his yom tov. Then there's erev Pesach, and Pesach, and isru chag when we don’t say tachanun. Since most of the month is holy we make the rest of the month holy [and we don’t say tachanun the entire month of Nisan]."
The Shlah HaKodesh expresses another reason we don't say tachanun in the month of Nisan. It is because each day in Nisan is like rosh chodesh. This is alluded to in the pasuk (Shemos 12:2), חדשים ראש לכם הזה החודש, "This month is for you rosh chodoshim." The entire month is holy like rosh chodesh.
The Agra d’Kala (Pikudei) teaches (in the name of Rebbe Mendel of Riminov zt'l) that the first twelve days of Nisan represent the twelve months of the year. ניסן 'א represents Nisan, ניסן 'ב represents Iyar, ניסן 'ג represents Sivan, and so on.
The following story is renowned among the Chozeh's students (and is written in Agra d'Kalah, Pekudei): The first twelve days of Nisan, which corresponds to the twelve months of the year, the Chozeh of Lublin zt'l would perceive what would happen that year, and he would always make a note of it. On ניסן 'א he would write what would happen in Nisan. ניסן 'ב he would write what would happen that year in Iyar, ניסן 'ג he would write what would happen in Sivan, and so on. And obviously, his predictions were always correct. In the final year of his life, he wrote what he perceived with his ruach hakodesh would happen that year until the month Av, but he didn't write down what would happen after that. His students wondered about that. But then he was niftar that year in Av, and it was understood why he didn't write that month.
We must know that Nisan is a very holy time, and we should strive to use these days well, and to fill them with Torah, tefillah, and good deeds.
The Avnei Nezer zt’l writes in a letter (see Avnei Nezer, Orach Chaim 336): "Shalom to my honored friend, the renowned gaon and genius, Reb Chaim, who was previously the Rav of Moscow: You asked that I answer some of your questions regarding Pesach. That isn’t a small request because these are priceless days—each hour is like a day. Nevertheless, since you expressed your strong desire that I answer you and you wrote that it would be your simchas yom tov, I devoted some time to writing this letter..." In the Avnei Nezer’s words, הקשה כי אם יחשב ליום שעה הערך יקרי אלו ימים כי לשאול, "You asked me for something difficult because these days are extremely valuable. One hour is like a day."
The Avnei Nezer's son, the Shem MiShmuel zt’l, testified that in Nisan, "My father was, העליונים בעולמות והולך מסובב, walking about in the upper worlds." He was more in heaven than on earth (Avir HaRo’im 264). The Shem MiShmuel once heard his father say that he doesn’t need to sleep in Nisan due to the holiness of these days (Avir HaRo’im 299).
The entire month of Nisan is special, and rosh chodesh is especially important. Rebbe Tzaddok HaCohen zt’l (Pri Tzaddik, HaChodesh 6) writes, "The holiness of the entire month lies...in rosh chodesh."
