What is compassion?

Before we can develop the compassionate desire to free others from suffering, first we need to identify what suffering is, from our own perspective. Then by extension, we can feel empathy for others who are going through suffering, whatever it might be. These two steps are essential.
The Dalai Lama

Compassion is a combination of empathy, reason and courage. When we embody empathy, we are warm-hearted and well-meaning. When we sharpen our reason, we develop true intelligence and the help we offer becomes practical and direct. Not only do we wish to help, we also have the discernment to know how to help. When combined with the courage not to be perturbed by adverse circumstances, our compassion becomes formidable.

Compassion is not necessarily a feeling. It is an aspiration that others be free from suffering. Initially, this 'other' may be someone you know and love. This aspiration can become both more vast, to include many others, and also infused with a deeper understanding of, for example, impermanence. 

Compassion, like mindfulness, is not something that is either 'on' or 'off'. Just as mindfulness deepens, becoming more refined and vast with experience, so compassion has infinite degrees and can be cultivated beyond all limits.