Burnside was born in London, and attended St. John's and Pembroke Collegess at the University of Cambridge, where he was second wrangler in 1875. He lectured at Cambridge for the following 10 years, before being appointed professor of mathematics at the Royal Naval College in Greenwich. While this was a little outside the main centers of British mathematical research, Burnside remained a very active researcher, publishing more than 150 papers in his career.
Burnside's early work was in applied mathematics. This work was of sufficient distinction to merit his election as a fellow of the Royal Society in 1893, though it is little remembered today. Around the same time as his election his interests turned to study of finite groups. This was not a particularly popular subject in late 19th century Britain, and his work on it was underappreciated until some years later.
The central part of Burnside's group theory work was in the area of group representations, complementing and sometimes competing with the work of Frobenius, who had founded the subject in the 1880s. One of his most memorable contributions was the proof of the paqb theorem (that all groups of order the product of two prime powers are solvable).
In 1987 Burnside's classic work Theory of Groups of Finite Order was published. The second edition (published in 1911) was for many decades the standard work in the field, and is still useful today.
Burnside is also remembered for the formulation of Burnside's problem, and for Burnside's Lemma, though the latter should actually be attributed to Frobenius and Cauchy.
In addition to his mathematical work, Burnside was a noted rower; while he was a lecturer at Cambridge he also coached the crew team. In fact, his obituary in the Times of London took more interest in his athletic career, calling him "one of the best known Cambridge athletes of his day".