Neither the exact date of birth or the date of death of Andrew Ellison have survived. He attended the common schools. After a subsequent law degree in 1835 and its recent approval as a lawyer, he began to work in Georgetown in this profession. Between 1840 and 1843 he was a prosecutor in Brown County. At the same time he proposed as a member of the Democratic Party launched a political career. In 1846 he sat as an MP in the House of Representatives from Ohio.
In the congressional elections of 1852 Ellison was in the sixth electoral district of Ohio in the U.S. House of Representatives in Washington DC chosen, where he became the successor of Frederick Green on March 4, 1853. Since he has not been confirmed in 1854, he was able to complete only one term in Congress until March 3, 1855. This was marked by the events leading up to the Civil War. After the end of his time in the U.S. House of Representatives Ellison practiced as a lawyer again.