Hebrew months started with the “sighting” of the new moon.  The sighting did not occur at the exact time of the new moon.  But it occurred when they could see the thin sliver of a crescent moon, as pictured above, after the actual time of the new moon.  That is what started the Hebrew month.  It did not start at the time of the actual new moon but the sighting of the crescent moon.  Rambam detailed the exact process as explained by Roy Reinhold. 

“1. The new moon was cited in this way. An observor stood and watched the sun go down in the West. When the last part of the sun disappeared on the horizon, the observor turned 180 degrees and faced the East. If the new moon was visible at the point and time, then the new moon was declared. If it was not visible then the new moon had not yet occurred. This is clearly spelled out in ancient historical records in Hebrew and Aramaic and Rambam wrote extensively on this issue explaining exactly how it was done and citing his references.  2. Secondly, Rambam (Rabbi Moshe ben Maimonides) also calculated the exact amount the moon must be above the horizon to be visually visible in Jerusalem. Rambam stated that the moon altitude at true sunset had to be 11 degrees above the horizon to be visible (with a number of technical conditions).”

So as soon as the sun went down if they saw the crescent moon they would declare that day to be the start of the new month.  So for example if sundown was at 6:00 P.M. today and the crescent moon was observed at sundown then they would declare the next day, which would start just after sundown, as the first day of the month.  It is also interesting to note that the Hebrew word for “month” is also the word for “moon.”