Atonement in islamic sense, is to be upset for your sins, sincerely ask forgiveness from Allah, make promise never to do that sin again.

There's two type of sins in Islam. 1. Sin against Allah: like not praying salah, not doing fast. 2. Sin against his creation: Stealing from someone, killing.

For first type of sin, you are to seek forgiveness from Allah. Make istigfar. There's many verse in quran and hadith of prophet about how to seek forgiveness.

Our Lord, indeed we have heard a caller [i.e., Prophet Muḥammad (ﷺ)] calling to faith, [saying], 'Believe in your Lord,' and we have believed. Our Lord, so forgive us our sins and remove from us our misdeeds and cause us to die among the righteous. 3:193

[Mention] when the youths retreated to the cave and said, "Our Lord, grant us from Yourself mercy and prepare for us from our affair right guidance." 18:10

As for the second type of sin, you need to seek forgiveness from both Allah and the victim. Like if you steal from someone, then no matter how much you ask for forgiveness from Allah, the sin will not be forgiven until you return the stolen goods/equivalent to the owner.

And lastly there's another way to atone for your sins. Getting the issue judged by islamic court. From the time of prophet Mohammad pbuh to near the end of Ottoman caliphat, the court followed the rules of quran-sunnah to resolve the case. That also counts as atonement. But It's not viable option now since caliphate has been abolished and no muslim country properly follow the sharia laws.

And the sacrifice of prophet Abraham is not about atonement. It's about the will to sacrifice your most precious things in favor of Allah. He became father at an old age, so his love for child was limitless. Then Allah ordered him to sacrifice the thing he loves the most. After a while, he realised that his son Ismael is the most precious thing to him. So he took permission from his child and tried to sacrifice him. But a camel got sacrificed instead. So prophet Ibrahim passed one of the hardest test of Allah and since then this became a ritual for muslims. We sacrifice cows or camels instead of sons and It's a reminder to us that we are the slaves of Allah and we should be ready to sacrifice everything for the sake of Allah.