Rosh Hashanah is the Jewish New Year festival. "Rosh" means "head" and "HaShanah" is "The Year" - so literally "Head of the Year". See also Rosh Hodesh (head of the month). Judaism actually has 4 traditional New Years relating to different aspects, but Rosh Hashanah is the most well known. It is celebrated on the first and second days of the Hebrew month of Tishrei. Due to Judaism following a lunar calendar, the exact date in the Gregorian Calendar varies, and can be anything from early September until early October. The first day of Rosh Hashanah can be a Saturday, Monday, Tuesday or Thursday but not a Sunday, Wednesday or Friday.
Unlike many cultures, Rosh Hashanah isn't a huge party. It is a time for reflection on what you've done over the past year, and what you intend to do in the forthcoming year. It is the start of the Yamim Noraim - the "Days of Awe", the holiest days in the Jewish Year. This includes Rosh Hashanah and Yom Kippur (although not the following Festival of Sukkot which is on a slightly lower level of holiness, on a par with Pesach and Shavuot). However, there is no actual repentance done during the prayers on Rosh Hashanah - this is saved mainly for Yom Kippur itself (and the non-festival days in the run-up to Yom Kippur).
Rosh Hashanah is quite unique in that it's celebrated by Orthodox Jews in both Israel and around the world for 2 days (see Second day Yom Tov). Briefly, the Jewish months started when the new moon appeared in Jerusalem, and it could take up to a week for the news of the new month to be communicated (on horseback) to Jewish communities around the world. So in Israel, where it was nearer, they would know the exact date and could celebrate the festival for one day - but around the world, they'd count from the previous month as well to be safe, and celebrate 2 days to be sure one of them is right. However, with Rosh Hashanah being the only festival on the first day of the month, they'd celebrate it on two days even in Israel to be sure of gettin the right day. That said, many non-Orthodox communities celebrate 1 day of Rosh Hashanah.
The Synagogue services on Rosh Hashanah are much longer than the normal festival prayers, and are chanted to a special solemn tune (Nusach) that is only used on Rosh Hashanah and Yom Kippur. Men traditionally dress in white, or wear a special white garment over their suits (a Kittel) and a white Tallit, white being a sign of purity. The focal point of the services is the blowing of The Shofar, a Ram's horn. There are many people - colloquially known as "3 times a year Jews" - who come to Synagogue only on the two days of Rosh Hashanah and on Yom Kippur. There are also well known key prayers such as Unetaneh Tokef and the "Aleinu" prayer, also said every day, where we proclaim our complete subservience to G-D. This prayer, when said on Rosh Hashanah, is one of the only times in the year that Jews kneel in Synagogue (in the Aleinu prayer on each day of Rosh Hashanah, and again on Yom Kippur together with 3 additional times).
As with any festival, there are festive meals on Rosh Hashanah - in the evening and at lunchtime. There is a special tradition of eating a piece of apple dipped in honey at the evening meals, a sign of a sweet new year (see Shanah Tovah). Also, many Jews put honey on bread at Shabbat meals for a month, instead of salt, as a sign of sweetness. On the other hand, many Jews won't eat nuts on Rosh Hashanah as the hebrew word for nut has the same numerical equivalent (gematria) as the word for sin.