In Hebrew, Rosh Chodesh means, literally, "head of the month" or "first of the month." Rosh Chodesh is the first day of any new month. If a month is 30 days long, then the 30th day is treated as part of the Rosh Chodesh for the next month, and the Rosh Chodesh for next month extends for two days (the 30th of the earlier month and the 1st of the later month).
In ancient times, Rosh Chodesh was a significant festival day. At that time, the new months were determined by observation. Each month began when the first sliver of moon became visible after the dark of the moon. Observers would watch the sky at night for any sign of the moon. If they saw the moon, they would report their sightings to the Sanhedrin, which would interrogate them to make sure that they were not mistaken. Where in the sky did the moon appear? Which direction was it pointing? If two independent, reliable eyewitnesses confirmed that the new moon had appeared and described it consistently, the Sanhedrin would declare the new month and send out messengers to tell people when the month began.
The day after the moon appeared was a festival, announced with the sounding of the shofar, commemorated with solemn convocations, family festivities and special sacrifices. The importance of this holiday in ancient times should not be underestimated. The entire calendar was dependent upon these declarations; without the declarations, there would be no way of knowing when holidays were supposed to occur.
In later days, however, the calendar was fixed by mathematical computation. After the destruction of the Temple, sacrifices were no longer available. Accordingly, the significance of this festival has substantially diminished. There are some slight changes to the liturgy for Rosh Chodesh, including the addition of part of Hallel after the Shemoneh Esrei, and some additional Torah readings, but that is about the only observance of Rosh Chodesh today.
It remains a custom in some communities for women to refrain from work on Rosh Chodesh, as a reward for their refusal to participate in the incident of the Golden Calf. See The Role of Women.
The Shabbat before Rosh Chodesh is known as Shabbat Mevarekhim, which means "the Sabbath of blessing." After the Torah reading in the Shabbat service, the prayer leader holds the Torah scroll, recites a blessing hoping for a good month, then announces the day of the upcoming week when the new month will begin and the name of the new month.
Shabbat Mevarekhim is not observed during the month of Elul to announce the beginning of the month of Tishri, the month in which Rosh Hashanah (the Jewish New Year) occurs. The common-sense explanation of this omission is simply that the month of Tishri is anticipated throughout the month of Elul with increasing intensity as Rosh Hashanah approaches, making a formal announcement of the date unnecessary. However, a Chasidic tradition teaches that G-d himself blesses the first of Tishri, the anniversary of Creation, and gave the privilege of blessing the rest of the months to the Jewish people.
Note that Shabbat Mevarekhim is not necessarily the last Shabbat of the month. In a 30-day month, the 30th is part of Rosh Chodesh for the next month. If the 30th falls on Shabbat, it is the last Shabbat of the month, but Shabbat Mevarekhim occurs on the 23rd, which is the last Shabbat before Rosh Chodesh.