Rosh Hashanah is the Day of Judgement for the entire world. On this day, a person is judged on his or her deeds, and whatever is to happen to him or her in the coming year is decided accordingly.

' "The eyes of G-D your G-d are on it continually from the beginning of the year until the end of the year" (Deuteronomy 11:12). What happens at the end of the year has already been decided from the beginning of the year (Rosh Hashanah)'. (Talmud, Rosh Hashanah 8a).

'On Rosh Hashanah every person in the world passes before G-d - they all pass in front of the creator one after the other.'
(Mishnah, Rosh Hashanah 1:2)

Yet, despite this, all are assessed simultaneously:

'"Who creates their minds together, who understands all their deeds" (Psalm 33:15),

means that G-d, Who creates everyone, sees into everyone's mind at the same time and understands all that each person does.'
(Talmud, Rosh Hashanah 18a)

The Torah instructs us to blow the shofar on Rosh Hashanah:

'You are to have a day of Sounding [of the ram's horn].'
(Numbers 29:1)

The main aspect of Rosh Hashanah is the recognition of G-d as Supreme King (i.e Supreme Judge) over the entire world. (The expression 'King' stems from the fact that the main function of a king used to be to rule as supreme judge over his kingdom).

Blowing the shofar symbolizes this:

'For G-d is ... the Great King over the world ... G-d has risen with a teruah, G-D [has risen] with the sound of the shofar.
(Psalm 47:3,6; recited in the Rosh Hashanah prayers immediately before the shofar is blown).

The shofar expresses the distinctive character of Rosh Hashanah as the Day of Judgement.

The phrases:

'You are to have a day of sounding' (Numbers 29:1)

and

'You are to have ... a remembrance of sounding' (Leviticus 23:23)

have been interpreted by the rabbis as referring to weekdays and Sabbath. On a weekday there is 'a day of sounding', but on Shabbat (when the shofar is not blown) there is 'a remembrance of sounding'.

Rosh Hashanah is celebrated on two days (the 1st and 2nd of Tishrei), although the Torah only mentions one day (Leviticus 23:24). The addition of the second day was a decree instituted by the early Neviim (Prophets) (Jerusalem Talmud, Eruvin 3:10).