There is an enormous amount of discussion on the issue all over the internet. My intent is to try and simplify the material (thanks to Martin Handwerker a regular attendee at my daf yomi shiur for his help in calculating).
There is a machlokes first recorded by the rambam (Kiddush Hachodesh chapter 9-10) how to calculate the solar year. Shmuel (eruvin 56a) simplifies the calculation of the year to 365 days 6hrs. However, R' Adda (commentaries on Rambam attribute this to R' Adda, but i do not know where it comes from in the Talmud) gives a more accurate calculation of 365d 5hrs 997chal 48reg = 365d 5hrs 55min 25sec (approx.).
The actual astronomical solar year is 365d 5hrs 48min 46sec.
Although R' Adda is more accurate in conforming with the astronomical calculation, the minhag was to simplify and follow shmuel even though we drift from the true equinox a little more than 5min per year. R' Moshe Feinstein (Igros Moshe O.C. 4:17, 5:17) responded to a suggestion to follow the R' Adda equinox by asking the questioner if he thought he knew more than all the Rishonim and Achronim who decided to follow shmuel.
The original calendar (Julian) also simplified the system and was totally consistent with Shmuel. Therefore, the secular equinox was on Sept. 21 or 22 (depending on the conversion of the extra 6hrs per year by making Feb. 29 days), so 60 days later was Nov. 21 or 22. However, there is a discrepancy of approx. 11 minutes per year between shmuel and the actual solar year which will cause the calendar to be ahead approx. 3 days for every 400 years.
In the 1500's Pope Gregory moved the calendar ahead by 11 days to compensate for the discrepancy of the past 1500+ years (11 min per year = approx. 11 days per 1500) and implemented a system to calculate for the extra 3 days per 400 years. The secular calendar changed from the Julian/Shmuel calendar to the "Gregorian" calendar. Therefore, in the time of Gregory they began Tal Umatar 11 days later than the Julian date - Dec. 2 or 3rd. Since then another 400 years have passed which makes the Gregorian calendar another 3 days ahead of the Julian/Shmuel calendar - so we ask for rain on the night of Dec. 5 or 6 which is the night after the day of Dec. 4 or 5.
Regarding the minhag to ask for rain 60 days after the equinox rather than earlier (7 cheshvon), R' Moshe cites Rashi 10a that we follow the custom of Bavel even though in other places we may have a greater need for rain (see Rosh).
