Short answers, pulled from the story.
Zazen is a meditative discipline and the primary practice of Zen Buddhism. Practitioners sit in a posture such as full-lotus or half-lotus with the spine erect, hands forming the cosmic mudra at navel level, and eyes half-lowered. Methods vary by school and teacher and include following the breath, repeating a koan or huatou, or practicing shikantaza, the open awareness approach central to the Sōtō school.
In the Rinzai school, zazen is typically combined with koan study, and practitioners sit facing each other with their backs to the wall. The Sōtō school makes less use of koans or none at all, preferring shikantaza, where the mind has no object; Sōtō practitioners sit facing the wall or a curtain.
Shikantaza is the approach central to Japanese Sōtō Zen in which the aim is just sitting, suspending all judgmental thinking and letting words, ideas, images, and thoughts pass without engagement. Practitioners take no specific object of meditation and remain as fully as possible in the present moment. Dogen described it in the Shobogenzo as thinking of not thinking, which he called nonthinking.
Hakuun Yasutani listed five kinds of zazen in his lectures for beginners as quoted by Philip Kapleau: bompu, which develops concentration to aid well-being; gedo, zazen-like practices from other religious traditions; shojo, small-vehicle practices; daijo, aimed at gaining insight into true nature; and saijojo, which corresponds to shikantaza.
The practice traces to early Chinese Buddhist sources, where the term zuòchán appears in the Dhyāna sutras. Kumārajīva, who lived from 344 to 413 CE, translated a text called the Zuòchán sān mēi jīng, a manual on the samadhi of sitting meditation. The Tiantai master Zhiyi, who lived from 538 to 597 CE, wrote influential works on sitting meditation that shaped later traditions.
Zazen is traditionally practiced in a meditation hall called a zendo, with practitioners sitting on a cushion called a zafu placed on a flat mat called a zabuton. Standard leg positions include full-lotus, half-lotus, the Burmese position, and seiza. Modern practitioners may use a chair with a wedge or cushion. The spine is held erect, the hands form the cosmic mudra, and breathing comes from the hara, the center of gravity in the belly.