A few general notes about Jewish holidays:
When Holidays Begin
All Jewish holidays begin the evening before the date specified. This is because a Jewish "day" begins and ends at sunset, rather than at midnight. If you read the story of creation in Genesis Ch. 1, you will notice that it says, "And there was evening, and there was morning, one day." From this, we infer that a day begins with evening, that is, sunset.
For a discussion of why Jewish holidays occur on different days every year, see Jewish Calendar.
Work on Holidays
Work is not permitted on Rosh Hashanah, Yom Kippur, the first and second days of Sukkot, Shemini Atzeret, Simkhat Torah, Shavu'ot, and the first, second, seventh and eighth days of Passover. The "work" prohibited on those holidays is the same as that prohibited on the Sabbath, except that cooking, baking, transferring fire and carrying, all of which are forbidden on Sabbaths, are permitted on holidays. When a holiday occurs on a Sabbath, the full Sabbath restrictions are observed.
Extra Day of Holidays
You may notice that the number of days of some holidays do not accord with what the Bible specifies. In most cases, we celebrate one more day than the Bible requires. There is an interesting reason for this additional day.
The Jewish calendar is lunar, with each month beginning on the new moon. The new months used to be determined by observation. When the new moon was observed, the Sanhedrin declared the beginning of a new month notice sent out messengers to tell people when the month began. People in distant communities could not always be notified of the new moon (and therefore, of the first day of the month), so they did not know the correct day to celebrate. They knew that the old month would be either 29 or 30 days, so if they didn't get notice of the new moon, they celebrated holidays on both possible days. For more information about the lunar months, see Jewish Calendar.
This practice of celebrating an extra day was maintained as a custom even after we adopted a precise mathematical calendar, because it was the custom of our ancestors. This extra day is not celebrated by Israelis, regardless of whether they are in Israel at the time of the holiday, because it is not the custom of their ancestors, but is celebrated by everybody else, even if they are visiting Israel at the time of the holiday.
Rosh Hashanah is celebrated as two days everywhere (in Israel and outside Israel), because it occurs on the first day of a month. Messengers were not dispatched on the holiday, so even people in Israel did not know whether a new moon had been observed, and everybody celebrated two days. The practice was also maintained as a custom after the mathematical calendar was adopted.
Yom Kippur is celebrated only one day everywhere, because extending the holiday's severe restrictions for a second day would cause an undue hardship.
Sources: Judaism 101