When Yom Kippur begins at Sundown on Friday, Jewish people around the world will begin to observe the holiest day of the year in their religion. For 25 hours — until Sundown on Saturday — they will refrain from any food and drink, including water. They will spend the majority of the day in synagogue, praying to God and repenting for their sins.
Unlike Rosh Hashanah, which comes just ten days before Yom Kippur, the day is not necessarily joyous and festive, but solemn and reflective. Consequently, it is not customary to greet those observing with "Happy Yom Kippur," although there are not necessarily repercussions if you do that. The best greeting to give to someone observing Yom Kippur in English is "have an easy fast." For those who are not fasting, but are observing the Yom Kippur, you can wish them a "Good Yuntif, or Yom Tov" which is Yiddish and Hebrew, respectively, for "Have a good holy day."
A more traditional greeting, and one specific to Yom Kippur, is "Gmar hatimah tovah" or "Gmar tov" which roughly translates into "a good seal." Those observing the holiday believe that the book of life, which determines an individuals fate for the coming year, opens on Rosh Hashanah, and is sealed at the end of Yom Kippur following the period of repentance.
Since Yom Kippur comes so close to Rosh Hashanah, you can say "Shana Tova," or "Happy New Year." Additionally, Yom Kippur falls on the Sabbath this year, so you can wish those in synagogue a "good shabbos" or "Shabbat Shalom."