Allyson Pimental has a good description of compassion and how it differs from sympathy and empathy, using a tale from the Buddha’s time about a woman who loses her only child, as well as personal experiences from her life. You can find her article here: Hands of Compassion
Jack Kornfield describes self-compassion in a short article, and identifies ways in which we can awaken self-compassion. You can find his article here: Awakening Self-Compassion
Kristin Neff is widely known for her teachings on self-compassion and practices that we can use to open ourselves to self-compassion. You can find an introduction to her approach to self-compassion here: What is Self Compassion?