Rosh Hashanah is about both the beginning of the world and its ending. It marks the point in creation where Adam was created by God, and is thus a time to begin (again). However, because it is also about God making a judgement on the person for the coming year, there is a sense of the final Day of Judgement too. So it is a serious time. Rosh Hashanah falls on the first and second of the month of Tishrei.
The time is announced a month in advance with the blowing of the Shofar (ram's horn) every day for a month before synagogue services except on the Sabbath. The sound is to motivate towards checking one's behaviour and becoming serious. There is first a long uninterrupted blast (tekiah), the second blast is three broken sounds (shevarim), the third has nine staccato (jerky) sounds (teruah) and the final uninterrupted blast (tekiah) is very long.
Rosh Hashanah is celebrated on two days. The tenth day of the season is Yom Kippur, when the focus is on sin and restoring oneself and thus getting closer to God. In between is the Fast of Gedaliah and Shabbat Shuvah, which is about Israel's (the people) return to God to receive forgiveness and prosperity. Rosh Hashanah itself focuses on God and what God may do.
On the first evening people may be dressed in new clothes, and they may eat a round challah loaf that symbolises the unbroken cycle of life. Apples are an alternative for their shape. Honey is eaten to represent a sweet new year. Fish with its head on (or just a head) is for good deeds to multiply like fish in the sea. Pomegranates are said to have 613 seeds, the same number of commandments for a Jew. Next day Taslich is putting fluff into a river to represent losing sins.
The liturgy says: Remember us unto life, for You, O King, delight in life: inscribe us in the Book of Life, for Your sake, O God of Life. Jews greet each other, "May you be inscribed and sealed for a good year!" In other words, that the God who knows everything about everyone puts a good entry into the Book for that year for that person.