Atonement (at-one-ment) in English is not the same as kippur in Hebrew, now generally understood as repentance. Both express facets of the same reality that prophets, priests, rabbis, and many others have struggled to discover, to explain, and to get the rest of us to do.
Yom Kippur יוֹםכִּפּוּר is described as a Day of Atonement or a Day of Repentance. Originally the word kippur comes from the root kopher כֹּ֫פֶר, meaning a ransom or other payment for favors received or expected, and referred to the practice of sin-offerings on behalf of all Jews at the Temple in Jerusalem. Judaism has come a long way since then, so that the primary meaning is no longer about transactions with God, but a change in one's heart.
Yom Kippur, in its name as we understand it now, and in the prayers offered on the day, calls on all Jews to repent, to apologize for past wrongs and vow not to do that again. Atonement is a different idea, originally at-one-ment in English. It can variously mean being at one with God, with all Jews, with oneself, with all of humanity, with the cosmos. It can be viewed as achieving oneness, or recognizing oneness that already exists. It is embodied in the phrase Tikkun ha-olam, healing the world entire, not just by social action, but by recognizing its original oneness and acting accordingly.
Neither atonement nor repentance mean punishing yourself for what you did. Nor do they mean perfecting the observances. They mean establishing yourself in righteousness far beyond observances. So how do you do that? Ah, well, on such matters, if you ask ten Jews, you get eleven opinions. (Don't get me started on what Christians try to turn Yom Kippur into.) I have given you hints of five major interpretations of atonement in two modes along with an extra, so there you go. Let's see what else Jews have come up with below the Orange Pillar of Cloud.