Beginning on Rosh Chodesh Nisan and continuing until the 13th of Nisan, we recite a series of verses (from Devarim 7) describing the offerings that the leaders of the 12 tribes brought for the inauguration of the Mishkan, the portable desert Sanctuary.
Each day we read the offering of a different tribe, ending with the lighting of the Menorah which represents the offering of the tribe of Levi.
Then a short prayer is recited in which we say, "...if Your servant is of the tribe... whose offering we read today in Your Torah, may all the holy sparks and holy lights that are included in the holiness of that tribe shine upon me, to give me understanding and intelligence in Your Torah and Your Fear, to fulfill Your will all the days of my life...""
Even Kohanim and Leviim, who do not belong to any of the tribes that brought the offerings, must read the Nesiim in the first days of Nisan, since it is something spiritual, related to souls, and there are situations in which a person has souls from different tribes within him, helping him in his service to God.