After many years of blissful ignorance, I finally sat down and worked out the mathematics involved, and I have added a page on The Jewish Calendar: A Closer Look, which may be of interest to those who want a deeper understanding or who want to write a Jewish calendar computer program.
For the rest of us, there are plenty of easily accessible computer programs that will calculate the Jewish calendar for more than a millennium to come. Note that the number of days between Nissan and Tishri is always the same.
In ancient times, the new months used to be determined by observation.
When people observed the new moon, they would notify the Sanhedrin.
Holidays are celebrated on the same day of the Jewish calendar every year, but the Jewish year is not the same length as a solar year on the civil calendar used by most of the western world, so the date shifts on the civil calendar.