While there are dictionaries that will give you the definition I think of compassion as something that means different things to different people. It is a feeling that one has for another human being but it doesn't mean that they are not held accountable for their actions. You can hold someone accountable while having a feeling of sadness for them or perhaps feeling a little empathy because there for the grace of (  ) go I. 
Deep inside every human being is hope, just sometimes people have never had it unless extended a hand or shown that it is there. Some just lose it in the depths of their addiction or mental illness.
I like the simple version of compassion meaning you consider others as human beings with with human failings that can strike any of us depending on our resilience to the world's woes.