The Torah spells out that this is a festival for the Jewish people. Rosh Hodesh is the festival for the beginning of Hebrew Calendar months, which always start on a New Moon.
"And on your joyous occasions - your fixed festivals and new moon days - you shall sound the trumpets over your burnt offerings and your sacrifices of well-being." (Numbers 10:10a)
The new month.
All work is allowed on Rosh Chodesh, however some women have a custom to not do certain types of housework on Rosh Chodesh. Some have the custom to not have haircuts nor cut their nails on Rosh Chodesh.
Rosh Chodesh (ראש חודש) does not have a specific Hebrew greeting.
Rosh Hashanah is on Rosh Chodesh. What you mean to ask is, What are the days between Rosh Hashanah and Yom Kippur. These are the Aseret Yemai Teshuvah, the Ten Days of Repentance.
Chodesh Tov means "have a good month" and is the traditional greeting used on Rosh Chodesh.
Rosh Chodesh refers to the new moon which traditionally marked the beginning of each month in the Jewish calendar.
There isn't any traditional Rosh Chodesh greeting for anyone, male or female. You could say "rosh chodesh same'ach" (ראש חודש שמח), but this sounds funny and strange.
It is important to know when Rosh Chodesh (the first of the month) occurs so that the Jewish holidays are celebrate at their correct times.
Rosh Chodesh is the first day of the new month on the Hebrew Calendar.
This year - 2009/5769 - Rosh Chodesh Adar is on 24th and 25th of February.
One of the songs written by the late Debbie Friedman z'l "Seasons of the Moon" is directly designed for Rosh Chodesh. It is on the compilation Songs of the Spirit and I am looking up the date.
Rosh Chodesh, the celebration of the new moon, is considered the 'women's holiday'.