They go into it simply here:
Once the date of the Molad Tishrei has been calculated, some additional considerations must be taken into account to determine the actual date of Rosh HaShanah. These considerations may cause the actual date of 1 Tishrei to be postponed from the date of the Molad Tishrei. There are four such postponements:
First, if the time of the Molad Tishrei is later than 18 hours from the beginning of the day, Rosh HaShanah is postponed to the next day. This probably accounts for the fact that the young moon could not have been observed until the next day.
Second, for common years only, if the Molad Tishrei falls on a Tuesday, and is later than 9 hours and 204 halakhim from the start of the day, Rosh HaShanah is postponed to the next day. This rule prevents a situation in which the postponements for the next year would require the year to be 356 days long.
Third, for years following leap years only, if the Molad Tishrei falls on a Monday, and is later than 15 hours and 589 halakhim from the start of the day, Rosh HaShanah is postponed to the next day. This rule prevents a situation that would require the previous year to be only 382 days long.
Finally, if Rosh HaShanah would fall on Sunday, Wednesday, or Friday, it is postponed to the next day. In combination with one of the above postponements, Rosh HaShanah could be postponed by as much as two days. This postponement prevents certain holidays from falling on the Sabbath.
The site also has a neat number crunching program that tells you when each of the above situations affect a given year's schedule and the rescheduled date for Rosh Hashanah.
Anyway, so the first three issues are essentially practical concerns to keep the entire system from going off kilter, but what really interests me is the last means of postponement. See, what it is referring to is the postponement rule of Lo ADU Rosh which is a mnemonic name, aleph, dalet, vuv to indicate that if the molad falls on the first, fourth or sixth day of the week then Rosh Hashanah is to be delayed. But why those days of the week? What's the significance?
So the Talmud says (Rosh Hashanah 20a) that the reason for the Wednesday and Friday rule is so that Yom Kippur doesn't fall out on either side of Shabbos and it's pushed off from Sunday because otherwise Hoshanah Rabbah can fall out on Shabbos (Sukkah 43a). So the Gemaras continue, respectively, explaining that having Shabbos and Yom Kippur after one another is bad either because then the dead may have to wait two days before burial and/or that food will go bad and that if Hoshanah Rabbah falls on Shabbos then people will be unable to perform the classic aravot beating.
Now, I find this really remarkable for a couple of reasons. The first reason is due to the fact that they seemed to think it was alright to mess around with the whole calender - and the dates of all the holidays - for what is essentially ritual convenience! In the pre-calculated system the molad would happen when the molad would happen and you couldn't just change around the dates at will. Yom Kippur no doubt _did_ fall out on Friday and Sunday on a regular basis. And even more than that, to ensure that people far from Jerusalem were observing the correct date for the holiday they kept two days yom tov - and yet here we are messing around with the holy times for convenience! It is really amazing.
Secondly, if the reason Yom Kippur was so moved was so that two days of Shabbos didn't follow one another - what would those Amoraim make of today's three days yom tov/Shabbos in a row?! But besides that, even within the calculated system, Pesach comes up against Shabbos pretty regularly - as it does this year - and Shavuous too regularly does the same. So I'm not even sure I understand their reasoning. Do the dead not have to wait the same two days? In any case, I'd much rather have it set up so that Pesach can never begin on a Saturday night rather than Yom Kippur getting that special concern.
Lastly, I'd like to say a few words about the molad itself. The molad was initially figured out by eye witnesses and was taken very seriously as the Mishnah and Gemara report in great detail. Indeed, witnesses could break the Shabbos to travel to the court to make sure that the molad was set on it's proper time. But as that system became impractical the molad was calculated into a calendric system that could operate independently of witness' sitings. So the molad was calculated to be 29 days 12 hours and 44+1/18 minutes after the previous molad (possibly taken from Ptolemy), which was actually very accurate. But the problem is that due to tidal effects between the Earth and Moon, the mean time for the Moon's orbit is now _less than_ what it was in Hillel II's day. The difference is about 0.6 seconds each month and the accumulated molad has by now gained almost 100 minutes on the actual molad that witnesses would report.
The consequence of this is that today the calculated molads are no longer somewhere between Israel and Babylon as it was during the Talmudic period. Now they are calculating a molad that happens somewhere over Afghanistan! And with a difference of over 100 minutes (out of 1440 minutes in day) that means that the molad is calculated to be on the _wrong day_ about 7% of the time! Link
Now that's some weird stuff, isn't it?
I guess I'll just leave this off with wishing all my readers a shana tovah umetukah and a pleasant Rosh Hashanah....assuming it actually is Rosh Hashanah, of course.