I
"Vayehi bayom ha'shemini" refers to the eighth day of the dedication of the Mishkan, Rosh Chodesh Nissan, when the Mishkan was established (Rashi, Vayikra 9:1). This day was the day on which Hashem's heart was joyful (Rashi Shir haShirim 3:11).
When Adar enters, we increase joy (Ta'anis 29a) commemorating the miracles of Purim and Pesach (Rashi). In a leap year, we read Shemini between Purim and Pesach, as we approach the midpoint between these two yomim tovim, - Rosh Chodesh Nissan - which is the day on which Hashem and Am Yisrael rejoiced.
How do we increase joy on Purim and Pesach? On Purim, we should increase gifts to the poor, "as there is no joy as great and splendid as gladdening the hearts of the poor, orphans, and strangers. One who gladdens the heart of these unfortunates resembles Hashem, "'Who revives the spirit of the lowly and the heart of the crushed'" (Rambam, Hilchos Megilla 2:17. Also see Purim: The Holiday of Giving). On Pesach, experiencing the authentic joy demanded by the mitzvah of "v'samachta b'chagecha" requires assisting the poor as well (Rambam, Hilchos Yom Tov 6:18). Rashi (Kiddushin 34b) goes further and reads v'samachta b'chagecha (Devarim 16:14) as v'seemachta, indicating that you must make others happy. Alternatively, only by making others happy can one himself be happy.
Despite the emphasis on achieving joy by helping the less fortunate, the primary obligation of joy on holidays is that a family be joyful together (Rambam, Hilchos Yom Tov 6:17, Ra'avad Chagiga 1:1). Furthermore, the requirement to be joyful as a family, "rejoice - you and your household" (Devarim 14:25), which is quoted in the context of the yomim tovim, applies year round as well (Tosafos Pesachim 109a). Indeed, one who dwells without a wife dwells without joy, as it says "Rejoice, you and your household" (Yevamot 62b).
II
"Vayehi bayom ha'shemini" - the word vayehi teaches that the joy was incomplete (Yalkut Shimoni 520) because on that very day, Nadav and Avihu died (Vayikra 10:2) as a punishment for bringing before Hashem a fire (aish zara) that He had not commanded them to bring (Vayikra 10:1).
Nadav and Avihu were great people, perhaps greater than Moshe and Aharon (Rashi 10:3). They were passionate in their love of Hashem and went so far as to pursue it without the limitation of the law by offering the ketores, which brings one extraordinarily close to Hashem. This, however, was against Hashem's will and led to their death (Ha'emek Davar 9:6, Harchev Davar 10:1).
Nadav and Avihu were unsatisfied with their exalted status and attempted to raise it in an unauthorized manner. Perhaps their unhappiness was connected to their not having established families of their own (as noted earlier from Yevamot 62b), as we are taught that they were punished for not having fulfilled this mitzvah (Yalkut Shimoni 524). This unhappiness led them to seek both spiritual heights which Hashem did not authorize them to experience as well as greater authority to rule over the people in place of Moshe and Aharon (ibid).
As the Rambam taught, gladdening others is a wonderful way to emulate Hashem and to achieve heights of joy which obviate the need for prohibited or inappropriate manifestations of religious fervor. This mandate is not limited to gladdening the poor on Purim and Pesach; as noted above, it applies within a family all year.
Raising children is the most important religious undertaking one can engage in (Igros Moshe 4:49). The satisfaction of serving Hashem in this critical manner, in a home which is imbued with the mandated simcha shel mitzvah discussed above, should protect against repeating the mistake of Nadav and Avihu who pursued a relationship with Hashem via an aish zara (halachically unauthorized religious activity) rather than through the authentic simcha shel mitzvah accessible through having and raising a family.
In these days of increased joy, may we all serve and cleave to Hashem appropriately, and by resembling and emulating Him, reach the greatest levels of simchah.