Pesach Sheni occurs every year on 14 Iyar. This is exactly one month after 14 Nisan, the day before Passover, which was the day prescribed for bringing the Korban Pesach in anticipation of that holiday. As described in the source text for this mitzvah, the Israelites were about to celebrate Passover one year after leaving Egypt. The offering of the Korban Pesach was at the core of that celebration. However "certain men" were ritually impure from contact with human corpses, and were therefore ineligible to participate in the Korban Pesach. Faced with the conflict of the requirement to participate in the Korban Pesach and their ineligibility due to impurity, they approached Moses and Aaron for instructions, which resulted in the communication of the law of Pesach Sheni.
As described in, a Jew may bring the Korban Pesach on Pesach Sheni if s/he was ritually impure due to contact with a dead body or was on a "distant journey". As elucidated in MishnahPesahim and its Gemara:
The definition of "distant journey" for this purpose is generally construed liberally.
All types of ritual impurity are generally included, not only those due to contact with a human body.
In net effect, anyone unavoidably prevented from offering the Korban Pesach on Passover may do so on Pesach Sheni. However one may not intentionally defer the Korban Pesach if it is in his/her power to offer it on Passover.
Similarities and differences compared to Passover
The Mishnah and Gemara outline the following similarities and differences in the laws of the Korban Pesach when it is offered on Pesach Sheni, compared to when it is offered on Passover: Same:
The particulars of the sacrifice itself—what animals can be used, that it must be roasted, that it must be eaten together with matzo and bitter herbs, that its bones cannot be broken, that it is eaten until midnight, that any leftovers must be burned.
Hallel is recited during its offering, offering overrides Shabbat prohibitions, only those previously "registered" prior to slaughter may partake.
Different:
The only restriction concerning chametz is that it cannot be eaten with the Korban Pesach. It is otherwise permissible to own it, and to see it in plain sunlight, unlike on Passover.
More generally, the full seven-day Ḥag haMatzot —the holiday now commonly referred to as Passover—is not observed in conjunction with Pesach Sheni.
Hallel is not recited at the meal when the Korban Pesach is consumed, unlike on Passover.
Due to the destruction of the Temple in Jerusalem, Jews are unable to perform the Passover sacrifice, neither on Passover nor on Pesach Sheni. Some have the custom to eat matzo during Pesach Sheni in memory of the sacrifice. Others say that the reason for eating Matzo is because the 14th of Iyar was the day that the Matzo that was taken out of Egypt was finished. Some have the custom to omit Tahanun from the daily prayer service, though this custom is not observed universally.
In Hasidism
Pesach Sheni is relatively prominent in Chabad as well as in Hasidic thought. One theme commonly expressed for this holiday is second chances. In several Hasidic groups, the rebbe conducts a tish on Pesach Sheni. The Rebbes of Nadvorna and related groups conduct a tish with four cups of wine, matza and maror, in the manner of a seder.
