Short answers, pulled from the story.
Anapanasati means mindfulness of breathing and Gautama Buddha attributed this form of meditation to himself. He described the practice in several suttas including the Anapanasati Sutta found in the Majjhima Nikaya which is numbered 118 within that collection.
The Buddhist monk An Shigao translated a version of the Anapana Smi Sutra between 148 and 170 CE after coming from Northwest India to China. The original translation was rediscovered at Amanosan Kongō-ji in Osaka, Japan by Professor Ochiai Toshinori on the 2nd of May 1999.
Abhidhamma literature discerns sixteen stages or contemplations of anapana-sati divided into four tetrads or groups of four steps. These stages progress through focusing on breathing as the body-conditioner followed by feelings mind itself and mental qualities.
The largest Tibetan lineage the Gelug came to view anapana-smi as a mere preparatory practice useful for settling the mind but nothing more while the living application has largely been lost. In contrast mindfulness based on anapana-smi is considered quite profound in other traditions particularly the Kagyu and Nyingma where it serves as the ideal way for the meditator to transition into taking the mind itself as the object of meditation.
It has been scientifically demonstrated that anapana-sati enhances connectivity in the brain and the brain grows in response to meditation. Meditators experienced in focused attention meditation showed a decrease in habitual responding in a 20-minute Stroop test according to Richard Davidson and colleagues.