To atone is to do something "right" to make up for doing something wrong. Religious believers are known to atone for their sins, but even students can atone for a past failure by acing a quiz or two.
The word atone came to English as a contraction of the words at and one. The verb means to make amends or reparations for an offense or wrong doing. You can remember the meaning by thinking of it as a sort of Zen concept, to be "at one" or in harmony with someone, you have to atone for your mistakes and be forgiven. In a religious sense, it means to repent for sins ("to atone for his sins").