Yom Kippur is ten days after Yom Teruah.  This is the Day of Atonement, or Judgment Day.  I will be honest and say that this isn't necessarily a fun holiday.  We're actually told to afflict ourselves on this day.  It is a day for repentance and introspection. 

“And it shall be a statute to you forever that in the seventh month, on the tenth day of the month, you shall afflict yourselves and shall do no work, either the native or the stranger who sojourns among you.  For on this day shall atonement be made for you to cleanse you. You shall be clean before the LORD from all your sins."
Leviticus 16:29-30

"Now on the tenth day of this seventh month is the Day of Atonement. It shall be for you a time of holy convocation, and you shall afflict yourselves and present a food offering to the LORD. 
Leviticus 23:27

"It shall be to you a Sabbath of solemn rest, and you shall afflict yourselves. On the ninth day of the month beginning at evening, from evening to evening shall you keep your Sabbath.”
Leviticus 23:32

"On the tenth day of this seventh month you shall have a holy convocation and afflict yourselves. You shall do no work" 
Numbers 29:7

"Then you shall sound the loud trumpet on the tenth day of the seventh month. On the Day of Atonement you shall sound the trumpet throughout all your land."  (This is for years of Jubilee.  Pretty cool that the Jubilee is announced on Yom Kippur - Clean, Atoned, and FREE!)
Leviticus 25:9