For most of us, when this happens, it's unexpected, maybe while walking on a mountain trail on a crisp autumn day, or being so focused in work or play that you are not thinking about past or future, or connecting with someone in a way that makes it seem like time is standing still. This state of being alive and whole in the present moment is always available, but it typically eludes us, especially in times of difficulty and external pressures.
"Mindfulness" is used in many contexts nowadays and there are many different understandings of the term. Diana Winston of UCLA's Mindful Awareness Research Center gives my favorite definition: Paying attention to present moment experience with open curiosity and a willingness to be with what is.
Mindfulness-Based Stress Reduction (MBSR) is a blend of meditation, body awareness, and mindful movement: learning through practice and study how your body handles (and can resolve) stress neurologically.