Question
When did the Jewish calendar start and why? Was it a special day?
Answer
You are asking a very good question. The calendar was actually the first commandment given to the Jewish people just before they left Egypt. G-d told Moshe that this month (Nissan) will be the first month of the year, and explained to him how to calculate the new months based on the lunar cycle (based on Exodus 6 and Talmud).
The names of the months came much later, they were brought back from Babylon with the exiles several hundred years later. Until that time they were known as the 1st month, 2nd month etc.
Rabbi David Sedley