“Mindfulness” of course can be broadly applied to any effort to pay intimate attention to present experience. The particular term “mindfulness meditation” is most often associated with Vipassana Buddhism, and especially in the ways folks like Jon Kabat-Zinn have adopted meditation practice for stress reduction, including work with chronic pain, and depression. Kabat-Zinn was trained, I believe, by Insight Meditation Society teachers (e.g., Joseph Goldstein and Jack Kornfield), who in turn have developed Insight or “Vipassana” Buddhism for the West out of their training in Southeast Asian Theravadin Buddhist monasteries.
That sort of mindfulness meditation is in essence a practice of being present and attentively, non-judgmentally aware of whatever phenomena most predominate in your conscious experience. Most commonly, one is encouraged to cultivate that quality of awareness through connection with attention to a fixed object, such as the sensations associated with breathing in and out. But ultimately, that basic capacity for being appreciatively present to your experience is the foundation for insight, seeing, for instance, how all our phenomenal experience arises and falls without inherent permanance or a substantive “selfhood” and without consituting a basis for ultimate satisfaction.
Vipassana/ Theravada is “old school” Buddhism, historically situated primarily in South Asia (Sri Lanka, Thailand, Myanmar). Zen koans and shikantaza, of course, are practices explored in the Zen school, a current within the East Asian Mahayana movement originating in China and subsequently spreading to Korea, Japan, and Vietnam.
There were classically “Five Houses” of Zen (or Ch’an, as Zen is called in China). Two of those five are now exant in Japan, namely, the Soto and Rinzai Zen schools.
Shikantaza is associated most with the Soto teachings. It can be translated as “just sitting.” It is ultimately a kind of choiceless presence, a “just being,” in and as any experience that arises whatsoever. It is akin to the Western contemplative practice of “coming to rest in a naked, blind, feeling of being” (The Cloud of Unknowing/ Epistle of Privy Counsel).
Many Soto Zen Buddhists will be acquainted with counting breaths, but in shikantaza, the spirit is more of letting go of the count and just following the physical sensation of breathing, and then letting go of even that level of directed focus, and practicing “seated enlightenment,” as the Japanese Zen Master, Dogen, might put it.
Koan practice is associated with the Rinzai Zen school. Early on, Ch’an/ Zen chose not to identify itself exclusively with any particular Buddhist scripture. Rather, it relied on a “direct-pointing to mind.” Stories of enlightened interactions between teachers and monks became favorite teaching devices, and these eventually became used as meditation objects, the “test cases” or, in Japanese, “koans.”
Even in intensive mindfulness or shikantaza practices, it is common sometimes to have archetypal psycho-emotional processes activated. In some respects, koans are in effect devices to jump-start that kind of contemplatively cathartic work. Someone given a koan typically has some years of experience already with mindfulness practice. The koan is brought to mind and/or “dropped” into one’s otherwise basically still meditation and, through close work with a teacher, one’s entire person is called upon to effect a “resolution” of the koan.