Just in case you were wondering, the appropriate greeting is L'Shanah Tova, which is shortened from Leshana Tova Tikoseiv Vesichoseim, Hebrew for "May you immediately be inscribed and sealed for a Good Year and for a Good and Peaceful Life".
If you didn't know, Rosh Hashanah marks the beginning of the new year in the Jewish Calendar, celebrated on the first and second days of the seventh month of Tishri (go here for more information). The holiday marks the last harvest of the season, or the great and final time of ingathering for the season. Today it's referred to among both Christians and Jews as the Feast of Trumpets. A BYU religion professor recently blogged about the holiday here, and mentioned the following:
One of the most festive of all Jewish holidays, Rosh Hashannah is one of the two High Holy Days in Judaism, the other being the solemn fasting day known as Yom Kippur (the Day of Atonement), which falls ten days afterward. The period between these two High Holy Days is known as “the Days of Awe,” a period when the Jewish faithful consider their trespasses, personal and national, reflect upon the need for repentance, and consider the future. It is significant that such a period is commenced with a festival as joyful and full of hope and anticipation for the future as Rosh Hashannah.Anciently, the holiday had a different meaning and was referred to as a day of remembering, when the Jews were commanded to blow trumpets on this day for remembrance - to remember their bondage and captivity, and also to plead that God would remember to keep his covenants and for Israel to remember their God.
Where this gets really interesting for latter day saints is when considering the statue of Moroni blowing his trumpet placed at the top of each temple. Where did this idea of him blowing a trumpet come from? You might want to answer by referring to Revelations 14:2 which says, "And I heard a voice from heaven, as the voice of many waters, and as the voice of a great thunder: and I heard the voice of harpers harping with their harps." But nowhere in that verse does it mention him blowing a trumpet. Well the answer has to do specifically with this holiday, Rosh Hashanah, or the Feast of Trumpets.
For several years, from 1823-1827 Joseph Smith had been preparing himself for his eventual mission of restoring the gospel of Jesus Christ to the earth, just as it was in ancient times. Part of this preparation included yearly visits from the angel Moroni. After several several years, Moroni visited Joseph Smith on September 22, 1827 to deliver to him the golden plates that contained what we now have in the Book of Mormon. That day also happened to mark the first day of Tishri, or The Feast of Trumpets. The cool thing is that you can search for a Jewish calendar converter, put in the dates and see that it corresponds. In fact, try it out here.
Moroni blows his trumpet from atop our temples because he came to officially mark the final gathering of souls before the second coming of Christ; he sits atop the temples because that is where we go to make the covenants that help us to remember God, and we hope will keep him in reminded of the promises he has made to us as his children. Not too long ago there was an article in the Ensign that goes into much more depth about this subject, here. It also mentions:
The blowing of the trumpet is the major ritual of the Feast of Trumpets. Because the first mention of the trumpet is at Mount Sinai, these instruments are seen by Jewish writers as a symbol of revelation (see Ex. 19:16, 19). The trumpet sound is therefore understood by them as a memorial of the revelation and covenant given on Mount Sinai. Yet Rosh Hashanah’s trumpet blasts have been accepted by many Jews not just as a memorial of the ancient covenant revealed at Sinai but as a prelude to a new and future covenant to be revealed, one that would result in Israel’s ultimate redemption. The day’s ritual includes a prayer regarding revelation named “Trumpets.”So as you continue to celebrate the Jewish New Year, and go through the next several Days of Awe until Yom Kippur (which is this Sunday the 27th), remember your Heavenly Father, and be aware that He remembers you.
Was the coming forth of the Book of Mormon on the Feast of Trumpets coincidental? Latter-day Saints who know about these events do not think so. Scriptural and prophetic truth is often manifest through fulfillment. The golden plates were delivered to the young Prophet Joseph Smith early in the morning of 22 September 1827. The Feast of Trumpets, with prayers pleading for God’s remembrance of his still-exiled people, had begun at sundown the previous evening. The services continued that morning, with a worldwide sounding of the ram’s horn. Unbeknown to Judah, all that those horns represented was now to be fulfilled. For on that day, God remembered His people and set in motion His plan to regather them. On that day, God’s final harvest began. On that day, new revelation was granted which would bring a return to renewed covenants. From that day onward, Israel would be called to repentance in preparation for Christ’s return and reign. The Book of Mormon exists to serve these ends. Today, Moroni’s image trumpets from temple spires around the world a final call to awaken, repent, and prepare.