Woods was born in Newark, Ohio. He attended college at both Western Reserve University and Yale University, graduating from Yale in 1845. Upon his graduation he returned home to Newark and studied law, being admitted to the bar in 1847 and establishing a practice with his tutor.

Woods, a loyal Democrat, was elected mayor of Newark in 1856, and to the Ohio General Assembly in 1858, being named Speaker of the House shortly thereafter. He opposed the Civil War but, not being a proponent of slavery, came to see the necessity of a Union victory. In 1862 he left the Ohio state house and joined the Union Army, where he fought at the battles of Shiloh and Vicksburg, rising to the rank of Major General before leaving the Army in 1866.

At the end of the war, Woods stayed in the South, settling in Bentonville, Alabama, where he reopened his law practice and began farming cotton. In 1869 he was named by President Grant to the U.S. Court of Appeals for the Fifth Circuit.

Woods sat on the Fifth Circuit for 11 years, before being named by Rutherford B. Hayes to the Supreme Court in December, 1880. He was the first person named to the high court from a Confederate state since 1853, though, being a northerner and {by that time} a Republican he was palatable to the Republican majority in the Senate.

Woods was not a major contributor to the court, spending only six years on the bench. He remained on the court until his death in 1887.