The Hebrew year is a lunar year. It consists of 12 or 13 months of 29 or 30 days each, with the 13th month being added to every second or third year as needed to keep the years aligned with the seasons. The length of the year alternates between roughly 354 days for a 12-month year and roughly 384 days for a 13-month year.
The beginning of a month was identified by sighting of the new moon, so this is what determined whether a given month was 29 or 30 days in length. The year was most often 12 months long, but a 13th month fairly often had to be added. How exactly they decided whether to add this 13th month isn’t precisely known – but we do know that for the sake of the yearly Passover feast, the years must have been kept fairly well synchronized with the seasons. The Passover feast, always beginning on day 14 of the first month, was very much tied to the springtime grain harvests. If they had too many 12-month years in a row, the Passover would begin coming too early to have any grain to harvest. Or if they added a 13th month too often they’d have had the reverse problem. Passover would come too late, and though the fields would be ready to harvest, they wouldn’t be allowed to begin the harvest until they had given the firstfruits offering, which happened right after Passover.
A consistent way to identify roughly when springtime begins is by paying attention to the Spring Equinox. The Hebrew calendar of www.TorahCalendar.com uses a year break rule which results in the Equinox falling between day 15 of the last month of the previous year (near the full moon of that month) and day 15 of the 1st month (again near the full moon, which this time is the Passover moon). If the Equinox would fall after Passover, that’s taken as too early for the year to begin. So what would have been month 1 is taken instead as the previous year’s month 13, with month 1 beginning a month later. We don’t know how closely this reflects what was actually done in Biblical times – but because of the reconstructable astronomical basis for the calendar, we can know that we’ll always be close to the actual dates – not more than one off in identifying the month of the year, and probably not more than one or two off in identifying the day of the month.
 Most likely the first visible crescent following the new moon is what identified day 1 of a new month.