Descended from a Worcestershire family, some of whom had sat in Parliament, he was born to William Colles and Mary Anne Bates of Woodbroak, Co. Wexford. The family lived near Millmount, a townsland near Kilkenny, Ireland, where his father owned and managed his inheritance which was the extensive Black Quarry that produced the famous Black Kilkenny Marble. The father died when Colles was 6, but his mother took over the management of the quarry and managed to give her children a good education.
While at Kilkenny College, there was a flood in which a local physician's house was destroyed. Abraham found an anatomy book belonging to the doctor in a field and returned it to him. Sensing the young man's interest in medicine, the physician let Abraham keep the book. He went on to enroll in Trinity College, University of Dublin in 1790 and received the Licentiate Diploma of the Royal College of Surgeons in 1795. Abraham went on to study medicine at Edinburgh Medical School, receiving his M.D. degree in 1797. Afterwards, he lived in London for a short period, working with the famous surgeon Sir Astley Cooper in his dissections of the inguinal region.
Following his return to Dublin, in 1799, he was elected to the staff at Dr Steevens' Hospital where he served for the next 42 years. He was a well regarded surgeon and was elected as president of the Royal College of Surgeons in Ireland in 1802 at the age of only 28 years. In 1804, he was appointed Professor of anatomy, physiology and surgery at the college.
In 1811, he wrote an important treatise on surgical anatomy and some terms he introduced have survived in surgical nomenclature until today. He is remembered as a skilful surgeon and for his 1814 paper On the Fracture of the Carpal Extremity of the Radius; this injury continues to be known as Colles' fracture. This paper, describing distal radial fractures, was far ahead of its time, being published decades before x-rays came into use.
He also described the membranous layer of subcutaneous tissue of the perineum, which came to be known as Colles' fascia. He also extensively studied the inguinal ligament, which is sometimes called Colles' ligament. In 1837, he wrote "Practical observations on the venereal disease, and on the use of mercury" in which he introduced the hypothesis of maternal immunity of a syphilitic infant when the mother had not shown signs of the disease. Colles' principal textbook was the two-volume Lectures on the theory and practice of surgery. He is regarded as the first surgeon to successfully ligate the subclavian artery.
In tribute to his distinguished career, Professor Colles was awarded a baronetcy in 1839, which he refused.
He retired in 1841 and died on 16 November 1843, from gout. He was buried in Mount Jerome Cemetery, Dublin.
In 1807, he married Miss Sophia Cope. His son followed in his footsteps, being elected to the Chair of Anatomy in the Royal College in 1863. Another of his sons married Elizabeth Mayne, a niece of Robert James Graves.
His grandson was the eminent music critic and lexicographer Henry Cope (H. C.) Colles.