We missed it in the United States, but elsewhere in the world (particularly North Africa and the Middle East) they saw a total solar eclipse today. Thus, today was the new moon, the time when the moon is directly between the earth and the sun. (Of course, not every month has a solar eclipse.) According to our mathematical algorithm, tonight is Rosh Chodesh Nisan. In a nod to the old days when the new month was declared based on testimony of witnesses, Rosh Chodesh may be observed on or after the date of the molad (lunar conjunction), but never before. In fact, if the molad occurs any time after noon, then Rosh Chodesh must be delayed to the next day, because it takes 6 hours from the conjunction until the crescent moon is visible (according to Rosh Hashanah 20b).
Back in the day, would the Sanhedrin have accepted testimony about a total solar eclipse as evidence of the new moon? If so, then it wouldn't be necessary to wait 6 hours. Sadly, I'm almost certain that the answer is no, given all the sources that talk about the shape of the crescent moon and such.
It's strange. The rabbis were certainly aware of eclipses (Sukkah 29a has some bubbe meises for solar and lunar eclipses), and were certainly aware of the phases of the moon (duh), but I've seen no evidence that they were aware that eclipses are linked to the phases of the moon (solar eclipses always occur at the new moon, and lunar eclipses at the full moon). Counterexamples, anyone?
Happy New Year! According to Rosh Hashanah 1:1, the 1st of Nisan (tonight) is the new year for kings and festivals.
Kings: On official documents, years were identified as "the nth year of King ____". This number was incremented on 1 Nisan, regardless of when the king took office. For example, in the United States, the 1st year of King George lasted from 20 January 2001 to 1 Nisan 5761. (Actually, any time from 1 Nisan 5760 to 1 Nisan 5761 could be referred to as either "the 9th year of King Bill" or "the 1st year of King George".) The 2nd year of King George was from 1 Nisan 5761 to 2 Nisan 5762. We just finished the 6th year, and are starting the 7th (and antepenultimate, baruch hashem) year of King George.
Festivals: Deuteronomy 23:22 says that if you vow to bring an offering, don't procrastinate! There are different opinions in the Gemara about exactly what the time limit is before you're in violation of bal t'acheir (don't be late). One opinion is that you have one full festival cycle to get everything in, and that cycle begins in Nisan, with Pesach.
But that's not all! Rosh Hashanah 7a says that the 1st of Nisan is also the new year for months, leap years, shekalim, and (some say) renting houses.
Months: Nisan is "the first month", even though the years are incremented in Tishrei (the seventh month).
Leap years: If you didn't add an extra month (Adar) to make the year a leap year, then once the 1st of Nisan rolls around, you've missed your chance.
Shekalim: The 1st of Nisan is the beginning of the fiscal year in regard to the shekel donations to the Temple. The offerings for this year have to be bought out of the shekalim from this year, and the 1st of Nisan is when that flips over.
Renting houses: If a lease says "for this year" (rather than "for one year" or "for 12 months"), then the lease goes until the 1st of Nisan.