The Simple Response: Passover Is Easy to Miss
From a practical standpoint, Passover is the only holiday whose central observance can be easily missed due to circumstances beyond one’s control.
For example, with enough effort, one can almost always make sure to have access to a shofar for Rosh Hashanah, a lulav and etrog for Sukkot, etc.
On the other hand, in Temple Times, the centerpiece of Passover was eating the korban Pesach (Passover offering) with the matzah and the bitter herbs. In fact, the very name of the holiday, “Pesach” (translated as Passover), comes from the Passover offering.
Now, this could only be done in close proximity to the Temple in Jerusalem (or its forerunner, the Tabernacle), and one had to be in a state of ritual purity. Due to these logistical hurdles, it was entirely conceivable for someone to miss the Passover offering and therefore need to take advantage of a “round two.”
(This also explains why the Second Passover is only for those who missed eating the offering; it is not, however, for anyone who missed out on matzah, maror, reading the Haggadah, etc. to catch up.)
In fact, this is exactly how Pesach Sheni came to be. We read in the Torah that as the first anniversary of the Exodus approached and G‑d instructed the people of Israel to bring the Passover offering on the afternoon of the 14th of Nissan, “there were, however, certain persons who had become ritually impure through contact with a dead body, and could not, therefore, prepare the Passover offering on that day.
They approached Moses and Aaron . . . and they said: ‘. . . Why should we be deprived, and not be able to present G‑d’s offering in its time, amongst the children of Israel?’”1
In response to their plea, G‑d established the 14th of Iyar as the Second Passover (Pesach Sheni) for anyone who was unable to bring the offering at its appointed time in the previous month.
The Deeper Reason: Passover Is Our Birth
The Chassidic masters explain that the underlying theme of the Second Passover is the concept of teshuvah (“return”), and that it is never too late to correct our mistakes. To quote the Hayom Yom of the 14th of Iyar:
The theme of Pesach Sheini is that it is never too late. It is always possible to put things right. Even if one was ritually impure, or one was far away, and even in a case when this (impurity, etc.) was deliberate, nonetheless he can correct it.
The Lubavitcher Rebbe explains that this is uniquely tied to Passover, which marks the “birth of the Jewish nation.”2
You can miss many things (buses, exams, work meetings and even flights) and make them up later. But if you miss being born, you really missed the boat.
Therefore, G‑d, wanted every person, even those who purposely missed the first Passover, to have a second chance at Passover, teaching that it truly is never too late to correct our deficiencies, even the most fundamental ones.3
Nowadays, even though we cannot bring the Passover offering, the custom is to still celebrate the Second Passover by eating matzah—shmurah matzah, if possible—and by omitting Tachanun from the prayer services. When marking this day, we should take the clear lesson to heart that no matter how far we may go, or how impure we may have become, G‑d will pave the way for us if we sincerely want to make amends.

A second chance. Passover is unlike other Jewish holidays. It is the most widely observed Jewish holiday, one largely observed at home, and the only holiday where the Torah gives a second chance for those who missed its initial observance. Perhaps this is because it's a holiday that defines who we are through telling our story, giving us boundaries, and calling us to justice. Passover is the story of our liberation from captivity, from oppression to freedom. We recall what it was like to be viewed with fear, mistreated and given no rights. This memory calls us to stand up to bigotry and injustice. Rabbi Yosef Yitzchak Schneersohn (1880-1950), the 6th Lubavitcher Rebbe, taught that the true significance of the Second Passover is to teach us that it is never too late to rectify a past failing. He understood this passage of Torah to be referring not just to those on a physically "distant road" but those who had strayed from the path; he understood the words those who were "contaminated by death" to be a metaphor referring to those who felt dead inside and struggled to feel God's love. We have an obligation to make sure that others have a chance to be full participants in the rites of our community, both religious and secular. This means giving people a second chance, the essence of Passover Sheni. Passover Sheni, this year occurs on April 29th (the 14th of Iyar). On that day, let us think of those who are in need of a second chance. Let us think of those who, perhaps, have strayed from the path, who have been in captivity, who have been viewed with fear but now are a part of us. Let us turn our minds to those who are asking to participate in one of the fundamental rights of American society - the right to vote. This Passover Sheni, let's hear the call for justice for those who have served their debt to society and are in need of a second chance.

• De dagelijkse Tachanoen (smeekgebeden) worden niet gezegd. (Nederland wordt het wel gezegd).
• In sommige gemeenschappen (zoals Chabad) hebben ze het gebruik om matzes te eten (Nederland niet).

Twee moppen als afsluiter:
Waarom vieren Jekkiem/Nederlanders (West Europese Joden, waaronder Nederland valt) geen Pesach Sjenie?
Antwoord: Ze komen op tijd voor de eerste keer en hebben dus geen tweede Pesach nodig ;)
Waarom eten Jekkiem/Nederlanders geen matse op pesach sjenie?
Antwoord: Ze hebben met Pesach precies het aantal matzes ingekocht voor pesach en zullen nooit matses over houden :)