In the tefillos of Rosh Hashonah we assume that Adam ha'rishon was created on the first of Tishrei. That day at that time was a Friday which today is impossible; according to the calendar that we use, Rosh Hashonah can never fall out on a Friday. Even though that day was the sixth day of creation, we refer to Rosh Hashonah as ze ha'yom techilas ma'asecha - the beginning of the creation of the world, because the first five days were pre-historic since there was no human being there to notice anything.
In the tefillos of Rosh Hashonah we mention that on this day of Rosh Hashonah we should all recall what happened on that very first Rosh Hashonah. The midrash tells us, commenting on the posuk in Tehillim, "Hashem ori v'yishi" that "ori" is a reference to Rosh Hashonah and "yishi" is a reference to Yom Hakipurim. On that very day that Hashem created Adam ha'rishon, he granted him illumination by instructing him to observe the basic mitzvot that apply to all of mankind. The Talmud (Bava Metzia 83b) comments that the possuk in Borchi Nafshi, "Toshes choshech vihi layla" is a reference to olam ha'zeh. Many issues in this world are very unclear just like in the middle of dark night; there are mitzvos that people think are really aveiros and there are aveiros that people consider to be mitzvos (see the hakdama of Mesilas Yesharim). Without illumination granted by Hashem through the laws of the Torah, we will remain "in the dark". Rosh Hashonah for Adam ha'rishon was his day of kabbolas haTorah.
According to the tradition recorded in the midrash, Adam ha'rishon sinned on that very same first day that he was created. He was judged and punished on the same day and Hashem notified him that just as I have judged you on this day, so too will I judge your descendants in all future generations on this day. The fact that Rosh Hashonah is the yom ha'din is never mentioned in Tanach but is an oral tradition from Adam ha'rishon (D'rashas HaRamban L'Rosh Hashana).
The story of the "original sin" does not really play a significant role in Jewish theology. It was recorded in the Torah, however, to teach us certain aspects about sin that are relevant to all of us today.
On Rosh Ha'shonah and all year long we should take to heart the details of the original sin and realize that it simply does not make sense to violate the mitzvot of the Torah.
[1] Quoted by the Ramban in his sefer Milchamos Hashem at the begining of Maseches Chulin
[2] See Mesilas Yesharim regarding perishus
[3] As stated in the aforementioned medrash
[4] This explanation is offered by the Beis Halevi