In the Hebrew calendar, Rosh Hashanah starts on 1 Tishrei and ends at the close of 2 Tishrei.
The Torah describes the festival as a one-day observance on the first day of the seventh month, using names connected with remembrance and the sounding of the shofar. Rabbinic Jewish practice developed into a two-day observance, and Orthodox and Conservative communities generally keep both 1 and 2 Tishrei, including in Israel. The two days are sometimes treated halakhically as one extended observance.
The holiday begins at sundown before 1 Tishrei, because Hebrew calendar dates begin in the evening. It ends after the second day, at nightfall, when 3 Tishrei begins. That next date is normally the Fast of Gedalia, a minor fast commemorating the assassination of the governor Gedaliah.
The main source of confusion is the difference between the biblical one-day description and later Jewish practice. Reform communities vary: many North American congregations observe only the first day, while others follow the traditional two-day pattern. Thus, 2 Tishrei is the standard answer for the customary two-day observance.