The word, "mindfulness," is actually a translation of the Pali word, 'sati,' which occurs in early Buddhist texts. 'Sati,' can also be translated as "attention," "awareness," or "heedfulness."
According to Wikipedia;
"...mindfulness is the psychological process of bringing one's attention to experiences occurring in the present moment, which can be developed through the practice of meditation and other training..."
In a period of mindfulness meditation the participants exercise an appreciative awareness of their physical sensations first of all, so that when they then apply awareness to thoughts and feelings as well, those thoughts and feelings are experienced in a broader context which tends to bring more calm, contentment and objectivity. Practitioners are also encouraged to practise a mindful, embodied awareness as much as they can, spontaneously the rest of the day, thus applying the method, ideally, in all aspects of their lives; at home, work and leisure.
Following the pioneering work of Jon Kabat-Zinn, a professor at the Massachusetts Medical School, who created the Mindfulness Based Stress Reduction programme, many psychologists in western nations have testified to the effectiveness of mindfulness as an antidote to stress, anxiety, depression and other disorders and also, more affirmatively, as a valuable aid to happiness, well being, productivity and good communication skills.
In positive terms, the practice helps us to appreciate our basic experience of the mind, body and senses which helps us to cultivate a degree of contentment and emotional stability. The practice also tends to highlight, simultaneously, our unhealthy stress reactions like anxiety, anger, misunderstandings and so on, as they form, which means that we can see them for what they are and regain a more balanced perspective by learning to respond instead with curiosity, openness, empathy and care. We could perhaps think of mindfulness as a kind, accepting awareness.