The biblical calendar actually has two different "new year" points, depending on which one you mean:
Nisan 1 — the religious new year (spring)
Established in Exodus 12:2: "This month shall be your beginning of months." This refers to the month of the Exodus from Egypt.
Falls around the new moon closest to the spring equinox — typically March or April.
This is the reckoning used for counting festivals (Passover is Nisan 14) and for dating the reign of kings in the Hebrew Bible.
Tishrei 1 — the civil new year (autumn)
This became Rosh Hashanah, the "head of the year" most people associate with the Jewish New Year today.
Falls around September, roughly six months opposite Nisan.
Associated with creation, judgment, and the agricultural/harvest cycle — and it's also when the year number increments (e.g., 5786 → 5787).
How the date itself is set:
Each new month begins at the new moon (originally observed visually, later fixed by calculation).