Yom Kippur is considered the holiest day of the Jewish year. It was the one day that the High Priest was allowed to enter into the Holy of Holies in the Temple to make atonement for the nation. Leviticus 16 gives us the time, reason, and customs of this day. It is to be observed on the 10th day of the 7th month (Tishri), it is a day for making atonement through the High Priesthood, and it is to be observed with fasting ("afflict yourselves") and special sacrifices.
For us, the day is no longer one in which we believe atonement is made for us on that particular day, for our atonement comes through Messiah's death and resurrection. But it is a day of remembrance of the atonement made for us and also a day to intercede for our loved ones that they may also enter into a relationship with God through Messiah's atonement.
It is also a day of introspection which is aided by fasting. As we fast, we humble ourselves before the Lord and open our hearts to His Word. We repent of misdeeds and allow His Ruach (Spirit) to enliven us to live holy lives. We do not, however, do this in a spirit of needing to earn salvation nor with an attitude of fear, for in Messiah we have the perfect High Priest that makes holy all those who come to Him. Many Jews wear white to services on Yom Kippur as a symbol of being purified. Whether you decide to wear white or not, come in knowing that in Messiah, your sins are gone!
Leviticus 16:29-34
"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. It is a Sabbath of solemn rest to you, and you shall afflict yourselves; it is a statute forever. And the priest who is anointed and consecrated as priest in his father's place shall make atonement, wearing the holy linen garments. He shall make atonement for the holy sanctuary, and he shall make atonement for the tent of meeting and for the altar, and he shall make atonement for the priests and for all the people of the assembly. And this shall be a statute forever for you, that atonement may be made for the people of Israel once in the year because of all their sins."
Hebrews 7:23-28 (Tree of Life Bible) Now on the one hand, many have become kohanim (priests), who through death are prevented from continuing in office. But on the other hand, the One who does remain forever has a permanent priesthood. Therefore He is also able to save completely those who draw near to God through Him, always living to make intercession for them. For such a Kohen Gadol (High Priest) was fitting for us: holy, guiltless, undefiled, separated from sinners, and exalted above the heavens. He has no need to offer up sacrifices day by day like those other kohanim g’dolim (High Priests)—first for their own sins and then for the sins of the people. For when He offered up Himself, He did this once for all. For the Torah appoints as kohanim g’dolim men who have weakness; but the word of the oath, which came after the Torah, appoints a Son—made perfect forever.