...In the seventh month, on the first of the month, there shall be a sabbath for you, a remembrance with shofar blasts, a holy convocation. -Leviticus 16:24
Rosh Hashanah occurs on the first and second days of Tishri. In Hebrew, Rosh Hashanah means, literally, "head of the year" or "first of the year." Rosh Hashanah is commonly known as the Jewish New Year. This name is somewhat deceptive, because there is little similarity between Rosh Hashanah, one of the holiest days of the year, and the secular celebration of the new year on January 1st.
There is, however, one important similarity between the Jewish New Year and the secular one: Many use the secular New Year as a time to plan a better life, making "resolutions." Likewise, the Jewish New Year is a time to begin introspection, looking back at the mistakes of the past year and planning the changes to make in the new year.
The name "Rosh Hashanah" is not used in the Bible to discuss this holiday. The Bible refers to the holiday as Yom Ha-Zikkaron (the day of remembrance) or Yom Teruah (the day of the sounding of the shofar). The holiday is instituted in Leviticus 23:24-25.
The shofar is a ram's horn which is blown somewhat like a trumpet. One of the most important observances of this holiday is hearing the sounding of the shofar in the synagogue. The Bible gives no specific reason for this practice. One that has been suggested is that the shofar's sound is a call to repentance. In the mystical teachings the vibrations of the sounds affect our inner selves bringing healing and alignment with our soul's purpose.
Another popular observance during this holiday is eating apples dipped in honey, a symbol of our wish for a sweet new year.
Practiced usually in the afternoon of the first or second day of Rosh Hashanah is Tashlich ("casting off"). We walk to flowing water, such as a creek or river, and toss breadcrumbs into the river, symbolically casting off our wrong doings.
...In the seventh month, on the tenth day of the month, you shall afflict your souls, and you shall not do any work ... For on that day he shall provide atonement for you to cleanse you before Adonai. -Leviticus 16:29-30
Yom Kippur is probably the most important holiday of the Jewish year. Many Jews who do not observe any other Jewish custom will refrain from work, fast and/or attend synagogue services on this day. Yom Kippur occurs on the 10th day of Tishri. The holiday is instituted at Leviticus 23:26 et seq.
The name "Yom Kippur" means "Day of Atonement. " It is a day set aside to atone for the wrong doings and regrets of the past year, to ask for forgiveness from G-d, others and ourselves. We also give forgiveness to G-d, others and ourselves. There is a misconception that on Yom Kippur we are atoning for our "sins." However the the Hebrew word, "chet" actually means "missing the mark."
Yom Kippur is a complete Sabbath with no work to be performed on this day. We also fast for 25 hours beginning before sunset on the evening before Yom Kippur, Kol NIdrei, and ending after nightfall on the day of Yom Kippur. Fasting is meant to help us become closer to G-d this day. We physically cleanse our bodies as we spiritually cleanse our inner selves.