Origin
The Book of Exodus establishes the beginning of the Hebrew calendar:
"And the LORD spoke unto Moses and Aaron in the land of Egypt, saying: 'This month shall mark for you the beginning of the months; it shall be the first of the months of the year for you.'" (12:1-2)
In the Book of Numbers, God speaks of the celebration of the new moon to Moses:
"And on your joyous occasions - your fixed festivals and new moon days - you shall sound the trumpets over your burnt offerings and your sacrifices of well-being." (10:10)
In Psalm 81:3 both new and full moon are mentioned as a time of recognition by the Hebrews.
The occurrence of Rosh Chodesh was originally based on the testimony of witnesses observing the new moon. When two reliable witnesses appeared before the Sanhedrin, the day was declared as Rosh Chodesh, either making the month a full month or a defective, 29-day month. After declaring the new month, news of it would then be communicated throughout Israel and the diaspora.
At a later date, a custom was developed in which an additional day could be added to the month to ensure that certain holidays (such as Yom Kippur) did not fall on the days before or after Shabbat.
Read more about this topic: Rosh Chodesh
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