The younger brother of the politician, Arthur Balfour, he was born at Edinburgh in Scotland. He attended Harrow School, where he showed no outstanding ability. However, one of the masters, George Griffith, encouraged and aided him in the pursuit of natural science, a taste for which, especially geology, he had acquired from his mother. Entering Trinity College, Cambridge, in 1870, he was elected a natural science scholar of his college in the following year, and obtained second place in the Natural Science Tripos of December 1873.
A course of lectures on embryology, delivered by Sir Michael Foster in 1871, turned Balfour's attention to animal morphology. After the tripos, he was selected to occupy one of the two seats allocated to the University of Cambridge at the Naples zoological station. The research work which he began there contributed in an important degree to his election as a fellow of Trinity in 1874; and also gave him the material for a series of papers (published as a monograph in 1878) on the Elasmobranch fish, which threw new light on the development of several organs in the Vertebrates, in particular of the urn-genital and nervous systems. His next work was a large treatise, Comparative Embryology, in two volumes; the first, published in 1880, dealing with the Invertebrates, and the second (1881) with the Vertebrates. This book displayed a vigorous scientific imagination, controlled by a logical sense that rigidly distinguished between fact and hypothesis, and it quickly won wide recognition, both as an admirable digest of the numberless observations made with regard to the development of animals during the quarter of a century preceding its publication, and as a work of original research.
Balfour's reputation was now such that other universities became anxious to secure his services, and he was invited to succeed Professor George Rolleston at Oxford and Sir Wyvilie Thomson at Edinburgh. Although he was only a college lecturer, holding no official post in his university, he declined to leave Cambridge, and in the spring of 1882 the university instituted a special professorship of animal morphology for his benefit.
He never delivered a lecture as professor. In the first term after his appointment he was prevented from working by an attack of typhoid fever. Going to the Alps for his health, he was killed, probably on 19 July, 1882, attempting the ascent of the Aiguille Blanche, Mont Blanc, at that time unscaled. Besides being a brilliant morphologist, Balfour was an accomplished naturalist, and had he lived would probably have become a leading taxonomist.