Self-compassion is extending compassion to one's self in instances of perceived inadequacy, failure, or general suffering. Neff has defined self-compassion as being composed of three main components - self-kindness, common humanity, and mindfulness.
Self-kindness: Self-compassion entails being warm towards oneself when encountering pain and personal shortcomings, rather than ignoring them or hurting oneself with self-criticism.
Common humanity: Self-compassion also involves recognizing that suffering and personal failure is part of the shared human experience.
Mindfulness: Self-compassion requires taking a balanced approach to one's negative emotions so that feelings are neither suppressed nor exaggerated. Negative thoughts and emotions are observed with openness, so that they are held in mindful awareness. Mindfulness is a non-judgmental, receptive mind state in which individuals observe their thoughts and feelings as they are, without trying to suppress or deny them. Conversely, mindfulness requires that one not be "over-identified" with mental or emotional phenomena, so that one suffers aversive reactions. This latter type of response involves narrowly focusing and ruminating on one's negative emotions.
Much of the research conducted on self-compassion so far has used the Self-Compassion Scale, which measures the degree to which individuals display self-kindness against self-judgment, common humanity versus isolation, and mindfulness versus over-identification. Research indicates that self-compassionate individuals experience greater psychological health than those who lack self-compassion. For example, self-compassion is positively associated with life-satisfaction, wisdom, happiness, optimism, curiosity, learning goals, social connectedness, personal responsibility, and emotional resilience. At the same time, it is negatively associated with self-criticism, depression, anxiety, rumination, thought suppression, perfectionism, and disordered eating attitudes 
Although psychologists extolled the benefits of self-esteem for many years, recent research has exposed costs associated with the pursuit of high self-esteem, including narcissism, distorted self-perceptions, contingent and/or unstable self-worth, as well as anger and violence toward those who threaten the ego.
It appears that self-compassion offers the same mental health benefits as self-esteem, but with fewer of its drawbacks such as narcissism, ego-defensive anger, inaccurate self-perceptions, self-worth contingency, or social comparison.
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^Neff, K. D & Vonk, R. (submitted). Self-compassion versus self-esteem: Two different ways of relating to oneself. Manuscript submitted for publication.