From Rosh Chodesh Nisan on there is a minhag to read the “Neseim” from Parsha Naso. (Shlah beginning of Pesachim “Hachodesh”, Mishna Berura 429:8, Be’er Heitiv 429:6, Kitzur Shulchan Aruch 107:1) The Kav Hayosher in Chap. 88 goes into great detail about this. He says to give tzedaka before reading the Nasi, since tzedaka always helps ones tefilos. He says this benefits that whole shevet. There is also a “Yehi Ratzon” to say after the reading. Reb Tzvi Elimelech of Dinov (Bnei Yisaschar) in Igra Dekala (Pekudei) says in the name of his Rebbe, Reb Menachem Mendel of Rimanov, that each of the first 12 days of Nisan represent 1 month of the year and contain all the koach of that coming month. He says it is well known that the Chozeh of Lublin would write down what would happen in the coming year during the days of the Neseim, each day for the corresponding month. The year he died he stopped writing at the month of Av; he passed away on Tisha B’Av. We see from here that one should daven for all his needs for the coming months of the year on each representative day of the Neseim. There are those that have a minhag to read the Neseim from a Sefer Torah, but most read it by themselves from a Siddur or Chumash.
The first day’s Nasi is from Shevet Yehudah, Nachshon ben Aminadav.