The holiday originates from the Torah in the Book of Numbers Chapter 9 in the wake of Moses' announcement that the Passover sacrifice (Korban Pesach) can only be eaten on a specific date and only by those who are ritually pure. These caveats to performing the mitzvah sparked anger amongst people in the nation who had come in contact with the dead and were now considered impure and unable to bring the offering - 'We are unclean by the dead body of a man; wherefore are we to be kept back, so as not to bring the offering of HaShem in its appointed season among the children of Israel?' (Numbers 9:7).
God responds to the nation by declaring that anyone who is unable to bring the sacrifice – either due to ritual impurity or an inability to reach Jerusalem (the site of offering) during Passover – can instead make the sacrifice on the 14th of Iyar, a full month after Passover, and eat the paschal lamb with matzah and maror (bitter herbs). As it is written, 'If any man of you or of your generations shall be unclean by reason of a dead body, or be in a journey afar off, yet he shall keep the passover unto HaShem in the second month on the fourteenth day at dusk they shall keep it; they shall eat it with unleavened bread and bitter herbs.' (Numbers 9:10-11). The bringing of the Passover offering was considered so important that this is the only instance of a Torah commandment in which an official "make-up day" is established to ensure performance.
Today, Pesach Sheni is celebrated only symbolically, as the practice of bringing sacrifices has been discontinued since the destruction of the Second Temple. Instead, it is now customary to eat a piece of matzah on the holiday while tachanun (a daily supplications prayer not recited on holidays) is omitted from the prayer service to commemorate the joyous occasion from ancient times.