Compassion is a feeling deep within ourselves a "quivering of the heart" and it is also a way of acting being affected by the suffering of others and moving on their behalf. Buddha and Jesus are the most well known exemplars of compassion, and it is the central ethical virtue in the two religions that developed from their teachings.
The spiritual practice of compassion is often likened to opening the heart. First, allow yourself to be feel the suffering in the world, including your own. Don't turn away from pain; move toward it with caring. Go into situations where people are hurting. Identify with your neighbors in their distress. Then expand the circle of your compassion to include other creatures, nature, and the inanimate world.
Why This Practice May Be For You
The practice of compassion increases our capacity to care. It reinforces charity, empathy, and sympathy. It is very good exercise for your heart muscle.
But when you move toward others with compassion, you are likely to bump into some common attitudes, just waiting to close your heart again. The usual suspects are judgment and all its associated "isms": racism, sexism, ageism, classism, and nationalism.
On a personal level, your compassion is sabotaged by feelings of ill will toward others: spite and malice. These feelings, and others arising out of emotional wounds and personal pain, are actually symptoms indicating that you need to have compassion for yourself.