As mentioned before, compassion means "to suffer alongside." To put yourself in someone else's shoes. Consider their perspective. Feel their pain. Jesus told some powerful stories to demonstrate what compassion looks like in real life, including the Parable of the Good Samaritan and the Parable of the Prodigal Son. Jesus, of course, also modeled this compassion consistently in his own life.
Matthew wrote of him, "When Jesus saw the crowds of people, he had compassion for them, because they were harassed and helpless, like sheep without a shepherd." (Matthew 9:36)
As Jesus met people from all walks of life (socially, ethnically, economically, politically, religiously, etc.) he was able to see them as individuals, enter into their situation, and speak grace and truth.
Many people tried to hijack Jesus' movement and say "He is one of us" and in doing so imply "He is against you." Jesus would have none of it. He constantly broke through their understandings. He pierced through their perceptions about who God is and who He might love. He did that for EVERY party. And he did it with compassion. He did it with the desire to see all of those groups learn to love the 'other.' And because there was a lot of legitimate bad blood between these groups, it would require a lot of forgiveness.
When Jesus came on the scene there were 4 major parties in the Jewish world- the Pharisees, the Sadducees, the Essenes and the Zealots. The Pharisees were self-righteous and legalistic, stacking law upon law on top of the already robust Old Testament. They hated those that didn't believe or behave like they did. Their power was in the synagogues which were scattered all over the Empire. Sadducees had sold out to the Roman Empire for political power and wealth. Their power was in the Temple, and they kept their power as long as they were able to keep the people subdued. Zealots were what you'd expect, violent types that wanted to overthrow Rome by force. Essenes abandoned society and tried to create their own societies. These four groups didn't like each other all that much. They blamed each other for their current predicament (enslaved to Roman Empire) and all thought that when the Messiah came he would agree with them that they were right and the others were wrong. Jesus baffled all of them by loving all of them. But even more scandalously, He loved the Romans too.
One of the most frustrating things Jesus said was "Love your enemies." It's right there in the middle of his biggest speech and you can't really get around it if you want to claim any sort of connection to him. He follows it up with, "Everyone loves people that love them. Everyone loves people that agree with them. How would that make you any different at all?"
As we look at the different stories of Jesus, we see that he backed it up. The self-righteous bigot who looked down on everyone? Jesus loved them. The religious leader who sold out his own people for political power and money? Jesus loved them too. The outcast and oppressed and silenced and ignored? Over and over and over again. Jesus fought for them especially it seems. But his love for one group did not come at the expense of extending love to another group.
His love is frustratingly consistent. Even more so his demand that if we are going to be a part of what he is about, we have to show that same, consistent love across the board.