Elias Nelson Conway was brother to James Sevier Conway, the first governor of Arkansas. He and his family moved to Missouri. Conway attended Bonne Femme Academy in Boone County, Missouri.
In 1833 Conway moved to Little Rock, Arkansas. He studied surveying and served as the state auditor from 1835 to 1849. In 1844 Conway was offered, but declined, the nomination for Governor in 1844. In 1853 the nomination was offered again and Conway accepted. His election campaign was successful and became Governor of Arkansas.
The Conway administration focused on physical improvement of the state. He formed the Chancery Courts and eased the state's financial problems. When Conway left office in 1860 and the state treasury held a surplus.
Conway retired from public life, becoming a bit of a recluse. Conway is buried at the historic Mount Holly Cemetery in Little Rock, Arkansas.
Conway was first cousin to Senator Ambrose Hundley Sevier and Governor Henry Massey Rector.