Basically sincere regret or remorse. Somebody usually turns to God or their beliefs to cleanse or dispose of whatever they consider to be remorseful. The film explores it in the sense that Kim Ja-hong died a hero's death. He was healthy and should have lived much longer, but he threw that away for a selfless, heroic effort. Before he can have the chance to be reincarnated, he is put on trial and his life is put under a microscope to see what demons Kim Ja-hong has kept hidden all these years.
If you're asking what repentance means in every day life, then it's really up to the individual and how religious he or she is. I think people try to live by the do unto others mentality if they have a regretful past and have the desire to be a better person in the present. But again, all of this is speculation and probably differs from person to person.