Short answers, pulled from the story.
Ernst Mach is best known for his work on supersonic shock waves and for the Mach number, which expresses the ratio of the speed of a fluid or object to the local speed of sound. He and physicist-photographer Peter Salcher presented their foundational paper on this subject in 1887, experimentally confirming the conical shape of a projectile's shock wave.
Einstein cited Mach's critique of Newton's concepts of absolute space and time as one of the three principles underlying general relativity. In 1930, Einstein wrote that it is justified to consider Mach the precursor of the general theory of relativity, and stated that he had read Mach's work with eagerness and admiration shortly before discovering relativity theory.
Mach rejected the atomic theory of matter on the grounds that atoms could not be directly observed. After a lecture by Ludwig Boltzmann in 1897, he reportedly declared that he did not believe atoms exist. His position followed from his broader phenomenalist philosophy, which held that only sensations are real.
In 1873, Mach independently discovered, at the same time as Josef Breuer, that the sense of balance is managed by information the brain receives from fluid movement in the semicircular canals of the inner ear. Physiologist Friedrich Goltz had identified the semicircular canals as relevant in 1870, but Mach and Breuer worked out the actual mechanism.
Mach bands are an optical illusion named after Ernst Mach in which the human visual system exaggerates the contrast between edges where slightly different shades of gray meet. The effect is caused by edge-detection processing in the brain and was among Mach's contributions to the science of sensory perception.
In 1901, Mach accepted appointment to the Austrian House of Lords but declined a hereditary noble title, judging it inappropriate for a scientist. He maintained personal ties to the Social Democrat politician Viktor Adler and left money in his will to the Social Democrat newspaper Arbeiter-Zeitung.