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Born in Eccles near Manchester, Chapman entered Manchester University in 1904 to study engineering. After graduating in 1907, his interest was diverted into more strictly mathematical areas, and he went to Cambridge to study mathematics, graduating in 1910. His first post was as chief assistant at the Royal Observatory, Greenwich, and his work there sparked off his lasting interest in a number of fields of applied mathematics, notably geomagnetism. In 1914 Chapman returned to Cambridge as a lecturer in mathematics, and in 1919 he moved back to Manchester as professor of mathematics, remaining there for five years. From 1924 to 1946 he was professor of mathematics at Imperial College, London. After working at the War Office during World War II he moved to Oxford to take up the Sedleian Chair in natural philosophy, from which he retired in 1953. However, his retirement meant no lessening in his teaching and research activity, which continued for many years at the Geophysical Institute, Alaska, and at the High Altitude Observatory at Boulder, Colorado.

The two main topics of Chapman's mathematical work were the kinetic theory of gases and geomagnetism. In the 19th century James Clerk Maxwell and Ludwig Boltzmann had put forward ideas about the properties of gases as determined by the motion of the molecules of the gas. Chapman's work, which he began in 1911, was the next major step in the development of a full mathematical treatment of the kinetic theory. The Swedish mathematician Enskog had been working, independently of Chapman, along similar lines, and the resulting theory is now generally known as the Chapman–Enskog theory of gases. While working in 1917 on mixtures of gases Chapman predicted the phenomenon of gaseous thermal diffusion. His subsequent work on the upper atmosphere was a practical application of his earlier more theoretical study of gases.

Highlights of Chapman's work on geomagnetism are his work on the variations in the Earth's magnetic field in periods of a lunar day (27.3 days) and its submultiples. This he showed to be the result of a small tidal movement set up in the Earth's atmosphere by the Moon. He also developed, in 1930, in collaboration with one of his students, what has become known as the Chapman–Ferraro theory of magnetic storms. In collaboration with Julius Bartels, Chapman wrote Geomagnetism (2 vols. 1940), which soon established itself as a standard work.