Today is Yom Kuppur, the Day of Atonement, for those who follow the Mosaic system. Information may be found in Leviticus 16 (NET Bible) in the Old Testament (Hebrew Bible). If you would like to read a modern Jewish description of the day read Judaism 101.
Christians who follow the New Testament believe that the Mosaic system has been removed and superseded by the gospel of Christ. Various New Testament writers have spoken on these matters. Here are a few significant references: Ephesians 2:14-16; Colossians 2:11-14; 2 Corinthians 3. Three entire New Testament epistles are devoted to discussions of the subject: Romans, Galatians, Hebrews. Hebrews 9:22-28 indicates that the offering of Christ is a one-time event, and sufficient to cover the sins of man.
Notice a few comments.
Hebrews 10:1-4 For the Law, since it has only a shadow of the good things to come and not the very form of things, can never, by the same sacrifices which they offer continually year by year, make perfect those who draw near. 2 Otherwise, would they not have ceased to be offered, because the worshipers, having once been cleansed, would no longer have had consciousness of sins? 3 But in those sacrifices there is a reminder of sins year by year. 4 For it is impossible for the blood of bulls and goats to take away sins (NASB).
Contrast this with the great proclamation of John the Baptist in the Gospel of John 1:29: “Behold, the Lamb of God who takes away the sin of the world”
Last year Todd Bolen posted a blog, with photos, of some residents of Jerusalem who “In the absence of a temple and the blood sacrifices that provide atonement for sins, some Jews today observe a sacrificial ceremony with a chicken.” You can find the full post here. I learned of a similar annual practice (not associated with Judaism) on a visit to Nepal a few years ago. Didn’t work before; won’t work now.