A few years ago, I was in a synagogue, and I overheard one man ask another, "When is Chanukkah this year?
" The other man smiled slyly and replied, "Same as always: the 25th of Kislev." This humorous comment makes an important point: the date of Jewish holidays does not change from year to year.
The year number on the Jewish calendar represents the number of years since creation, calculated by adding up the ages of people in the Bible back to the time of creation. D." means "the year of our L-rd," and we do not believe Jesus is the L-rd. The American "new year" starts in January, but the new "school year" starts in September, and many businesses have "fiscal years" that start at various times of the year.
The civil calendar used by most of the world has abandoned any correlation between the moon cycles and the month, arbitrarily setting the length of months to 28, 30 or 31 days.
This process is sometimes referred to as "fixing" Rosh Hashanah. However, the Jewish New Year is in Tishri, the seventh month, and that is when the year number is increased.
If you are interested in the details of how these calculations are performed, see The Jewish Calendar: A Closer Look. This concept of different starting points for a year is not as strange as it might seem at first glance.
In ancient times, the new months used to be determined by observation.
When people observed the new moon, they would notify the Sanhedrin.