Rosh Hashanah and Yom Kippur are the High Holy Days of the Biblical calendar.
While Rosh Hashanah is a joyous time, Yom Kippur is very serious. It is the Day of Atonement, when we seek forgiveness of our sins.
"The LORD said to Moses, 'The tenth day of this seventh month is the Day of Atonement. Hold a sacred assembly and deny yourselves, and present an offering made to the LORD by fire. Do no work on that day, because it is the Day of Atonement, when atonement is made for you before the LORD your God...You shall do no work at all. This is to be a lasting ordinance for the generations to come, wherever you live. It is a sabbath of rest for you, and you must deny yourselves. From the evening of the ninth day of the month until the following evening you are to observe your sabbath."
(Leviticus 23:26-28, 31-32, NIV)
The Bible teaches that sin separates us from God.
"Surely the arm of the LORD is not too short to save, nor His ear too dull to hear. But your iniquities have separated you from your God; your sins have hidden His face from you, so that He will not hear." (Isaiah 59:1-2, NIV)
During the Day of Atonement, the High Priest offered the blood of a sacrificed animal to make an atonement, or covering, for sins. "For the life of a creature is in the blood, and I have given it to you to make atonement for yourselves on the altar; it is the blood that makes atonement for one's life."
During Yom Kippur, Jews fast and pray, repenting of sin, and asking God's forgiveness. Since there is no longer a Temple or High Priest in Jerusalem, many hope that their good works and prayers will be enough to cover their sins. But God's requirement of a blood sacrifice has not changed. It has been fulfilled, however, in the atoning blood sacrifice of Jesus the Messiah!
"But now a righteousness from God, apart from law, has been made known, to which the Law and the Prophets testify. This righteousness from God comes through faith in Jesus Christ ["Christ" is the Gentile way to say "Messiah"] to all who believe. There is no difference, for all have sinned and fall short of the glory of God, and are justified freely by His grace through the redemption that came by Christ Jesus. God presented Him as a sacrifice of atonement, through faith in His blood."
(Romans 3:21-25, NIV)
If you would like more understanding of this, please read through the book of Hebrews in the New Testament.
Since our sins have been forgiven, what can we, as believers in Jesus the Messiah, do to observe Yom Kippur?
Prayer and fasting and rest are really powerful ways to get closer to God, not because we are required to do so to obtain forgiveness, but out of a desire to know Him and please Him more. Memorizing Scripture is another way to honor Him. A good verse for Yom Kippur is...
"If we confess our sins, He is faithful and just and will forgive us our sins and purify us from all unrighteousness."
1st John 1:9, NIV
Another thing you might want to do is plan a special meal after the fast, to celebrate God's forgiveness through Jesus the Messiah!