Shanah Tovah is a Hebrew greeting which means "Happy (Tovah) New Year (Shanah)".
Not surprisingly, this is most commonly used by Jews around the time of Rosh Hashanah, the festival of the New Year. It is appropriate to be used from a week or so before Rosh Hashanah, up to and including Yom Kippur which occurs about a week later. In many Synagogues, people will also simply say "Happy New Year" (or no doubt the equivalent in their own language!).
There are also a number of variations on the greeting. "Shanah Tovah U'mtukah" means "Happy and Sweet New Year" - the theme of a "Sweet New Year" runs through the festival - for example, it's traditional to eat apple dipped in honey at the evening festive meals, and for all the festivals in the month many people put honey on the bread after Kiddush rather than the usual salt.
Another variation is "Shanah Tovah Tikateivu (v'Teichateimu)". This means "Happy New Year, and you should be Written (and sealed)". This is based on the tradition that on Rosh Hashanah G-D writes everybody's name in the "Book of Life" or the "Book of Death" - but then he doesn't seal it until Yom Kippur (see the second half of Unetaneh Tokef). By saying this, you are implying they should be written and sealed in the Book of Life.
The past 2 days (4th / 5th October 2005) were Rosh Hashanah. Try using this greeting on your Jewish friends!