Sharon Salzberg explores compassion and compares it to several other ways of relating to suffering, both suffering in the world and our own suffering. Those include anger, fear and grief among others. She emphasizes that compassion is a practice that reminds us of our intention to respond with kindness, compassion, and wisdom in the presence of suffering. You can find her article here: The Nature of Compassion.
Pema Chodron has a good article on the Lions Roar website that gives a brief but complete introduction to tonglen, a meditation for the cultivation of compassion for ourselves and others. As she points out in the article, the practice of tonglen reverses several of the attitudes that are ingrained in our culture. That can have a profound effect, opening us to compassion for the world. You can find her article here: How to Practice Tonglen Meditation
Roshi Joan Halifax is abbot of Upaya Zen Center, and has taught tonglen to many students. In her experience, tonglen can help us to dissolve the armor that surrounds our hearts. She wrote a document that provides more information about the emotional setting of tonglen and gives several variations of the practice. You can find her article here: Tonglen