All of the "celebrations" surrounding the New Year made me ponder whether there's a particularly Jewish approach to calculating time. I know Rosh Hashanah is the Jewish New Year, but is there a Jewish number of years? Also, I was wondering whether there are Jewish months, or even Jewish days for that matter.
Yours is a very timely question.
According to the civil calendar in use now, called the Gregorian calendar, it is currently the Year 2013. This reckoning of years is based on the purported year of the birth of Jesus. It is for this reason that historically the years of this calendar were referred to as BC (before Christ, Greek for Savior) or AD (Anno Domini, Latin for "year of our Lord"). In modern times, in order to avoid the particular religious connotation of these terms, many customarily refer to these years as BCE (before the Common Era) or CE (the Common Era).
Aside from the obvious aversion to using a system of time that posits Jesus as Savior and Lord, Judaism has its own reckoning of years whose use pre-dates any other calendar and whose starting point reaches back the furthest. Based on the Torah's chronology of Creation and the stated years of the generations following Adam and Eve, the Jewish calendar is currently in the year 5773 from the year in which these progenitors of mankind were fashioned by G-d. The Jewish calendar, therefore, reckons the beginning of Creation for all mankind and marks the inception of the relationship between G-d and Man.
Since you mention the "celebrations" surrounding the New Year, it's worth pointing out the difference between the hedonistic way many celebrate New Year's (which, historically, often resulted in horrific attacks against Jews) as opposed to the Jewish observance of Rosh Hashanah that is imbued with repentance and sanctification as befitting the day celebrating the purpose of Creation.
To answer your second question: Yes, there are specific Jewish months as well. And here too, they are significantly different than the months of the Gregorian calendar. For one, the names and origins of many of these months are pagan: January=Janus, god of gates; March=Mars, god of war; April=Aphrodite goddess of love; May=Maia, goddess of spring; June=Juno, goddess of marriage and women. As above, Judaism has a natural aversion to using a system based on unacceptable beliefs.
Another major difference is that the Gregorian calendar is solely solar based, so while it is in sync with the seasons, there is no relationship between the months and the various phases of the moon. This is not the case in the Jewish calendar where the beginning of the month occurs on the new moon, the middle of the month occurs on the full moon and the month concludes at the end of the waning moon. Since there are approximately 29.5 days in the lunar cycle, some months are 29 days and others are 30, but all correspond to the phases of the moon.
However, a purely lunar-based year is also out of sync with the times. This is because twelve 29.5-day moon-months result in a 354-day year, which lags behind the 365-day solar year by 11 days. Over three years, a lunar calendar will lag behind the seasons by approximately a month, and throughout the years, its months will drift throughout the entire spectrum of the seasons. The Islamic calendar is such a system, where, for example, the month-long, day-time fast of Ramadan is sometimes in the short, cool days of winter and other times in the long, hot days of summer. The Islamic calendar is thus the converse of the Gregorian calendar; while its months are in sync with the moon, they are totally out of sync with the sun and the seasons.
Despite its preceding Christianity and Islam, the Jewish calendar ingeniously resolves the tensions that rise from their calendars with a soli-lunar calendar that simultaneously preserves within the seasons of the solar year true lunar months. The way this works is that since a discrepancy of about one month occurs over three years, approximately every three years (specifically, 7 times in 19 years) a second month of Adar is added at the end of the winter in order to fulfill the Torah's mandate for Nisan, the month of Redemption, to occur in the spring. The intercalation of this extra month ensures that all of the months and their holidays are always in balance with their respective seasons.
Regarding Jewish days, given the antiquity of the Torah, it would seem that the seven-day week delineated in the Torah is the basis for that being the universal standard. However, here again, the names of the days of the week in the Gregorian calendar have their origin in idolatry and paganism where each day is dedicated to a different planet/god: Sunday to the sun; Monday to the moon; Tuesday to Tew, the Norse equivalent of Mars; Wednesday to the god Weden; Thursday to Thor; Friday to Freya, the equivalent of Venus; and Saturday to Saturn.
The Jewish days of the week, based on the wording of the Torah, are simply numbered from one to six (corresponding to Sunday through Friday), as a count-up culminating in the only day of the week referred to by name - Shabbat. This is the day that is to be celebrated as the completion and perfection of Creation. In their attempt to transplant yet imitate Judaism, Christianity and Islam chose respectively the closest day after or before Shabbat as their Sabbath. But according to the source for their choice, the Torah, both days are mundane. Jewish sources describe this choice as a Divinely directed fulfillment of the Torah verse establishing the unique relationship between Israel, Shabbat and G-d.