On Yom Kippur, Jews say they're sorry. They say they're sorry to God for wrongs they've committed against God, and they also say they're sorry to friends, family, and acquaintances who they've wronged over the course of the year. If you've been mean to someone and hurt their feelings, you have to apologize to them and ask them for forgiveness, not God.
Jewish repentance, or teshuvah, is based on the idea of "returning." Since the Jewish concept of sin is about "going astray" or "missing the mark," it makes sense that the goal is to "return" to goodness and righteousness. There are three steps to teshuvah: acknowledging the wrong done, offering to correct the wrong, and promising never to wrong again.
These are simple yet profound steps that can easily be taught to young children. It's simply not enough to say "I'm sorry." Truly righting a wrong is hard work. It takes self-awareness, diligence, and commitment. But the rewards, both for the one giving the apology and the one receiving the apology, are so much more rewarding because of it.
May you and your children enjoy a year full of joy, learning, growing, and good fortune!