Napoleon Bonaparte Burrow was born in 1818 in Bedford County, Tennessee, the son of Banks Mitchum and Mary (Blanchard) Burrow. His father was a farmer in Bedford and Carroll counties. Entering Nashville University in 1836, the young Burrow graduated from the law department in 1839 and commenced a legal career. He settled in Huntingdon in Carroll County, and practiced law there until the outbreak of the Mexican War. A second lieutenant of the 2nd Tennessee Volunteers, Burrow fought with great distinction in Scott's assault on Mexico City. After the war he settled in Arkansas, first in Jefferson County and later near Van Buren. In both places he practiced law and was a substantial planter and slaveholder. Burrow was also active politically as a state senator from Jefferson County from 1851 to 1855, a Buchanan elector in 1856, and a delegate to both 1860 Democratic conventions. By January, 1860, Burrow was a general in command of a brigade of Arkansas militia.

A prominent secessionist, Burrow was the candidate of the ultra-secessionist "Hindman" faction to the First Confederate Senate. When Arkansas seceded, Burrow and his militia brigade (3rd Brigade, First Division) took over Fort Smith, Arkansas, from the federal garrison. His command was so criticized - one influential editor called his conduct there "extravagant and... pompously unmilitary" - that he was relieved after two weeks. The Arkansas Military Board later appointed Burrow a brigadier general and sent him to Springfield, Missouri, after the Battle of Wilson's Creek, to transfer the Arkansas militia there to regular Confederate command. The militiamen, encouraged by their commander, Brigadier General N. B. Pearce, refused to transfer, and Burrow's mission ended a fiasco. Burrow spent the rest of the war raising crops for the army. Despite his ill-fated last command, Arkansas politicians retained confidence in Burrow's abilities and recommended to President Davis that he be appointed a Confederate general, a recommendation that Davis, beset by numerous petitions of a like nature, never acted upon.

After the war Burrow resumed his farming and legal careers in Van Buren. In the latter he gained great notoriety, one contemporary stating that "as a criminal lawyer ... in the state, he stood at the head of his profession." General Burrow died of pneumonia at Alma in Crawford County, Arkansas, on May 23, 1880, while returning to Van Buren from a legal case. He is buried in the Alma City Cemetery.