"Mindfulness means paying attention in a particular way: on purpose, in the present moment, and nonjudgmentally." - Mindfulness teacher Dr. Jon Kabat-Zinn, Wherever You Go, There You Are (1994), p4.MBSR is very specific in it's definition of mindfulness.
This video by Kristin Neff, associate professor in human development and culture at the University of Texas, Austin; Kristin Neff: Mindfulness and Self-Compassion (from the "Practicing Mindfulness & Compassion" conference on March 8, 2013 co-hosted by The Greater Good Science Center and Mindful magazine) is interesting because of the way Neff breaks mindfulness down into layers as follows:
M - Mindfulness
m1: Present Moment
She states that each level is cumulative; that one cannot have m2 without m1 first being in place, and one cannot have m3 without m2 being in place, so this gives us a linear view of how mindfulness unfolds as we learn how to apply it.