Born in Aylsham, Norfolk and educated in Norwich, Clover was apprenticed to a local surgeon at the age of 16, and later entered University College Hospital, London. He qualified in 1847 and was appointed apothecary to the hospital where he operated, administered anaesthetics and taught students. In 1850 he became a Fellow of the Royal College of Surgeons, although most of his life was spent as an anaesthetist. He lived at 3 Cavendish Place, London where, in his own workshop, he made the prototypes of his many inventions, the most famous of which is his portable regulating ether inhaler.
In 1863 Clover led the chloroform enquiry of the Royal Medical and Chirurgical Society, which recommended that the pulse be monitored throughout an anaesthetic. He played a major part in the introduction of nitrous oxide during the late 1860s, and worked with the leading surgeons of the day.
Clover died of uraemia aged 57 and was buried in the Brompton Cemetery. He is regarded as one of the founders of British anaesthesia, and is a supporter on the coat of arms of the Royal College of Anaesthetists, which holds a biennial Clover lecture in his honour.