Rosh (head) Ha (the) Shanah (year). Why is it called the Head of the Year? Because the head rules the body. So the Rosh rules the Shanah. So why is the Jewish new year in the 7th month of the Jewish calendar? Because Pesach (Passover) is in the first month of the Jewish calendar. And Rosh Hashanah and the New Moon always coincide. The sighting of the new moon begins the Jewish month. Not the first day of the secular calendar, but usually somewhere near the middle of the secular month.
Rosh Hashanah is celebrated two days. If the first day falls on Saturday, the shofar is not blown, because the shofar is NEVER blown on the Sabbath. In that case it is blown on the second day.
We greet everyone with the phrase L'Shana Tova: A good year. We also dip apple slices in honey and eat other sweet things so that the coming year will be sweet in all things: life, profession, financial, love...everything.
But these two days aren't the end of the "season." Next comes Yom Kippur. During these ten days of Rosh Hashanah (a Day of Remembrance) to Yom Kippur (the Day of Judgment), we review the past year, our sins and our good deeds,
resolve to do better the coming year. We call this the The Ten Days of Repentance.
The standard greeting to others during this time is L'shanah tovah tikatevu vetehatemu (May you be inscribed and sealed for a good year). On Rosh Hashanah, we are inscribed (we hope "for good") in the Book of Life, and sealed On Yom Kippur.
Exodus 12 begins to explain why Passover is the first month of the Jewish year. Leviticus 23 ennumerates the various festivals of the year, and verse 23 specifies "a day of rest for you, a remembrance with shofar blasts, a holy convocation." And verse 26 ff. specifies Day of Atonement. That chapter continues to ennumerate the various holy days.
Any questions? As a rabbi.
Shalom