According to my handy Luach, on Pesach Sheini, some people omit tachanun (I haven't looked into any source texts to see who does) and according to the Wikipedia page, some eat matzah but I'm wondering about the nature of the day. Was there a sense of simcha for those people who were not observing Pesach Sheini through bringing a sacrifice in the time of the beit hamikdash? If not, why would anyone at all today mark the day in any way at all? And wouldn't those who "observe" it today through custom do so for 2 days because there would be the same concern over the date of Pesach Sheini outside of Israel as there would be for computing the date of Pesach itself!
In his Luach Hayom Yom, the Lubavitcher Rebbe explains the message of Pesach Sheini: "The theme of Pesach Sheini is that it is never too late. It is always possible to put things right. Even if one was tamei (ritually impure), or one was far away, and even in a case of "lachem", when this (impurity etc.) was deliberate - nonetheless he can correct it." This idea that we always have a another chance to fix things is good cause for celebration.
See here regarding ommiting Tachanun on the 15 Iyar.