Dublin-born Smith studied at Oxford and had a great interest in classics – it was only after a good deal of hesitation that he chose mathematics as a profession instead. He remained in Oxford in various capacities for most of the rest of his life. In 1860 he became Savilian Professor of Geometry there.

Smith's main work was in number theory and his greatest contribution was his development of a general theory of n indeterminates, which enabled him to establish results about the possibility of expressing positive integers as sums of five and seven squares. This achievement ought to have won Smith the prestigious prize offered in 1882 by the French Academy for their mathematical competition. However Smith, a notably unambitious man, did not enter and the prize was in fact given to Hermann Minkowski. When it was discovered that Minkowski had made use of crucial results published by Smith, the French Academy hastened to transfer the prize to Smith, but as he had unfortunately died in the meantime his fame was only posthumous. Smith also worked on the theory of elliptic functions.