Short answers, pulled from the story.
Samatha refers to calm or tranquility of the mind, developed through concentration on a single object, while vipassanā refers to insight or clear-seeing into the true nature of reality. In the Pali Canon they are described as two qualities of mind to be developed together, while later traditions sometimes treat them as separate techniques or even rival paths.
The modern Vipassanā movement was reinvented in Myanmar (Burma) in the 18th century by Medawi (1728-1816). The "New Burmese Method" was formally developed by U Nārada (1868-1955) and popularized by Mahasi Sayadaw (1904-1982) and Nyanaponika Thera (1901-1994). The movement began spreading to the West starting in the 1950s.
The three marks of existence that vipassanā seeks to reveal are anicca (impermanence), dukkha (suffering or unsatisfactoriness), and anattā (non-self). Insight into these three characteristics is the goal of vipassanā practice in the Theravāda tradition.
The nine stages, called the nine mental abidings, progress from initial placement of attention on a meditation object through continuous placement, repeated placement, close placement, taming, pacifying, fully pacifying, single-pointing, and balanced placement. The culminating state of śamatha proper is sometimes counted as a tenth stage, in which the meditator can maintain absorbed concentration for about four hours without interruption.
In the Mahamudra tradition, the practitioner does not actively construct tranquility but allows it to arise naturally by suspending value judgments on mental activity. Rather than following the nine-stage concentration model of the exoteric manuals, Mahamudra treats the ability to catch oneself in the act of distraction as the true test of tranquility meditation, and emphasizes equanimity toward both meditative equipoise and discursive thoughts.
The Visuddhimagga, a 5th-century Theravāda treatise by Buddhaghosa, is identified by Gunaratana as the classical source for treating samatha and vipassanā as two distinct meditation vehicles. Buddhaghosa describes a sequence in which jhāna is induced by samatha and then made the object of vipassanā; critics including Thanissaro Bhikkhu and Richard Gombrich argue this separation is not found in the earlier Pali suttas themselves.