In the tradition of our Sages
Bavli, Rosh Hashana, 15, 71
From the little that is said about the day, we learn that it is forbidden to perform any "labor" in it. A "Mikra Kodesh" - reading and proclamation of the sanctity of the day - was conducted on that day, and it was marked with a special sacrifice. It is also said to be a "Teruah remembrance" or a "day of a Teruah," as a King's fanfare, homage to the Lord who ascends to his throne to judge the land with justice and trial.
The scripture does not specify what instrument they used for the Teruah, but our Sages, bless their memory, learned from what is stated in Parashat Hayovel (Jubilee): "Then shalt thou make proclamation with the blast of the horn on the tenth day of the seventh month" (Leviticus 25: 9) "What is the shofar blowing there - even here the Teruah of the Shofar", and they Instituted Tekisah (blowing) and Teruah (proclaiming) of the Shofar on Rosh Hashana.
According to Chazal's tradition, Rosh Hashana is the Day of Judgment and it opens the Ten Days of Repentance that end on Yom Kippur:
“At four seasons judgement is passed on the world: at Passover in respect of produce, at Pentecost (Shavuot) in respect of fruit; At the holiday (= Sukkot) in respect of the water; and man is judged on Rosh Hashanah and his sentence is sealed on Yom Kippur”…