Quain, born at Fermoy, co. Cork, in July 1800, was third son of Richard Quain of Ratheahy, co. Cork, by his first wife. Jones Quain was his full brother, and Sir John Richard Quain was his half-brother. Richard received his early education at Adair's school at Fermoy, and, after serving an apprenticeship to a surgeon in Ireland, came to London to pursue the more scientific part of his professional studies at the Aldersgate School of Medicine, under the supervision of his brother Jones. He afterwards went to Paris, where he attended the lectures of Richard Bennett, a private lecturer on anatomy and an Irish friend of his father. In 1828, when Bennett was appointed a demonstrator of anatomy in the newly constituted school of the university of London (now University College) Quain assisted his patron in the duties of his new office. Bennett died in 1830, and Quain then became senior demonstrator of anatomy, Sir Charles Bell at that time occupying the professorial chair of general anatomy and physiology. When Bell resigned this post, Richard Quain was appointed professor of descriptive anatomy in 1832, Erasmus Wilson, Thomas Morton, Viner Ellis, and John Marshall successively acting as his demonstrators. He held the office until 1850.