Joseph Thomas Clover was born at Aylsham, Norfolk on 28 February, 1825. At the age of 16, he was apprenticed to Charles Mends Gibson, a surgeon and apothecary of the city of Norwich. As a pupil of Gibson, Clover was allowed to attend the Norfolk and Norwich Hospital, a mid-eighteenth century foundation with a first-class reputation. Before he became articled, Clover had already attended operations by Dr. Lubbock and Mr. Crosse of Norwich. From 1841-1845, he attended the Norwich Hospital, then from 30 September 1844 to 17 May 1845 he was at University College Hospital, London, attending upon Robert Liston, Richard Quain and other notable surgeons.

He became Resident Medical Officer at University College Hospital in 1848, and was admitted as a Fellow of the Royal College of Surgeons in 1850. Initially, Clover was drawn to the field of urology. He practiced as a surgeon, inventing two instruments for the crushing and removal of bladder stones. Ill health caused him to give up in 1853 and he turned to general practice. He set up his practice at 3 Cavendish Place, London, which became his home until his death in 1882. After several years in general practice he devoted his practice to anesthetics, and became "chloroformist" to the University College Hospital, the Westminster Hospital and the London Dental Hospital. Clover's choice of speciality helped to fill the vacancy created by the death of John Snow in 1858.

Considered an expert in anesthesia, Clover was sought out when important figures required surgery, administering chloroform to Napoleon III of France, on several occasions, Alexandra of Denmark, her husband Edward VII, Sir Robert Peel, Florence Nightingale and Sir Erasmus Wilson. Clover also invented and improved a myriad of anesthesia medical apparatus, including a chloroform apparatus and a portable regulating ether inhaler.

Clovert died of uraemia in 1882 and is buried in Brompton Cemetery, London, 200 yards away from John Snow.