What are the differences between the First Passover and Second Passover, Passover Sheini?
On the First Passover, [leaven] is forbidden to be seen or to be found [in one's possession]; on the Second Passover, leaven and matzoh coexist in one's home...
-Talmud, Pesachim 95a
A mitzvah is a commandment - G‑d instructing man what He desires for man to do or not do. Understandably, then, virtually all of the Torah's 613 mitzvot are unilateral declarations of divine will: one does not see many "proposals" for mitzvot being presented to G‑d, or "negotiations" between the supreme legislator and His earthly constituents.
One of the rare exceptions to this norm is the mitzvah of Pesach Sheini, the "Second Passover." The First Passover, as we all know, commences on the evening following the 14th of Nissan, the night that the Jewish people were liberated from Egypt. The Second Passover comes one month later, on the 14th of Iyar, and was instituted as a result of a petition by several individuals who were unable to participate in the First Passover.
At the heart of Passover is the korban Pesach (the "paschal lamb"), which was offered in the Holy Temple on the afternoon of the 14th of Nissan-indeed, all other observances of the festival (the eating of matzoh and maror, the prohibition against leaven), as well as the festival's very name, are related to the Passover offering.
The laws of the Passover offering mandate that only those who are in a state of taharah (ritual purity) may offer and partake of it. One year after the Exodus, as the Jewish people were preparing to celebrate their first Passover outside Egypt, a group of Jews approached Moses. They explained that they were ritually impure because they had come in contact with a corpse; the law would therefore preclude their bringing a Passover offering. But they refused to reconcile themselves to this. "Why should we be deprived?!" they cried; why should we be excluded from observing the festival of redemption, as will the entire community of Israel? G‑d responded by instituting a second Passover especially for those who, for whatever reason, were prevented from offering the Passover offering in its appointed time.