The mitzva of Rosh HaShana is the sounding of the shofar. What is the connection between the shofar and the Day of Judgment? The shofar spurs us to leave our evil ways and think about how to repent and do more good deeds.

The sounding of the shofar is a wake-up call to arouse us:

Even though the sounding of the shofar on Rosh HaShana is a [divinely legislated] Torah edict [for which there is no explicit reason], there is an allusion associated with it, as though it is saying: Awaken, sleepers, from your sleep; and slumberers, arise from your slumber, examine your actions and repent, and remember your Creator. Those [of you] who forget the truth due to the futilities of the time, and err all year long with emptiness, which provides neither benefit nor salvation: Look into your souls and improve your ways and your actions, and each one of you should abandon his evil path and his thoughts that are no good.

One is obligated to view himself as though he is being judged at every moment, and as though the two sides of the scale are equally balanced. An act in one direction or the other determines how the scale will tilt:

Therefore, every person needs to view himself all year long, as though he is half innocent and half guilty, and likewise, the entire world is half innocent and half guilty. If he has commited one sin, he has tilted the scale for himself and for the whole world in the direction of guilt, and has caused its destruction. If he has fulfilled one mitzva, he has tilted the scale for himself and for the whole world in the direction of innocence and has brought about salvation for himself and for them, as it is stated: “The righteous is the foundation of the world” (Proverbs 10:25); he who has acted righteously tilted the whole world to [a ruling of] innocence, and saved it. Because of this matter, the entire house of Israel customarily increases charity and good deeds and occupies itself with the performance of mitzvot from Rosh HaShana until Yom Kippur, more so than during the rest of the year. (Rambam, Mishneh Torah, Hilkhot Teshuva 3:4)

Further reading: For the halakhot of how and when to blow the shofar, see A Concise Guide to Halakha, p. 147.