How does the Jewish calendar work today?

The first mitzva given to the Jewish people is החדש הזה לכם the mitzva of קידוש החדש. This meant that Rosh Chodesh was determined by the testimony of the witnesses in front of the בית דין and that it was their determination and pronouncement that actually made it Rosh Chodesh. At some point about 1800 years ago this stopped and since then we have been using a fixed calendar. The obvious question is how does this work?

The Rambam states both in the Sefer Hamitzvos and the Mishna Torah that there are 2 systems in halacha of determining when Rosh Chodesh is.
1.קידוש החדש
2.A הלכה למשה מסיני of חשבון (calculation, e.g. the calender).

According to the Rambam when קידוש החדש stopped about 1800 years ago, the הלכה למשה מסיני of the calendar kicked in.

The Rambam writes that nowadays what is מקדש is the ניהוג of ראש חדש by the Jews in Israel. Hilchos Kiddush Hachodesh (5,13)
זה שאנו מחשבין בזמן הזה כל אחד ואחד בעירו ואומרין שראש חודש יום פלוני, ויום טוב ביום פלוני--לא בחשבון שלנו אנו קובעין ולא עליו אנו סומכין, שאין מעברין שנים וקובעין חודשים בחוצה לארץ; ואין אנו סומכין, אלא על חשבון בני ארץ ישראל וקביעתם. וזה שאנו מחשבין, לגלות הדבר בלבד: כיון שאנו יודעין שעל חשבון זה הן סומכין, אנו מחשבין לידע יום שקבעו בו בני ארץ ישראל איזה יום הוא; ובקביעת בני ארץ ישראל אותו הוא שיהיה ראש חודש או יום טוב, לא מפני חשבון שאנו מחשבין.

RYBS explains the Rambam as follows. He has a יסוד (with many proofs, not for now) that one of the jobs of the Sanhedrin is to represent the Jewish nation (this is for example why real semicha can only be given in Israel). Based on this he explains as follows. קידוש החדש is really a mitzva on all of the Jewish nation. However, all of the Jewish people cannot do קידוש החדש, therefore the Sanhedrin does it as our representative. However, when we switch over to the הלכה למשה מסיני of חשבון, the mitzva reverts back to the Jewish nation and it is the ניהוג of the Jewish nation that is what is actually מקדש, and because the Jewish people as a nation only exists in Israel (there are many proofs to this, not for now) it is the קביעות of the Jews living in Israel which is what is actually מקדש the חדש.

This Rambam is used to answer the question of why the Rambam left out the mitzva of ישוב הארץ?

Some answer that we see from this Rambam that ישוב הארץ is not just a regular mitzva, the whole calendar depends on jews living in Israel and therefore it is a mitzva כללית and according to the Rambam's principles is not listed.

In any case, we see the tremendous emphasis the Rambam placed on living in Israel.

The Ramban disagrees with the Rambam and claims that he never heard of such a הלכה למשה מסיני, rather, he explains that the calendar is really based on the regular halacha of קידוש החדש. The chachamim at that time got together and were actually מקדש all the future ראשי חדשים, and they left us a calendar which tells us what days they were מקדש.