He studied theology, but owing to a throat affliction, he abandoned his studies in that area and began the study of medicine. He graduated as a physician from the College of Philadelphia in 1768, receiving the first medical diploma issued on the American continent.
In 1769, Archer commenced the practice of law in Harford County, Maryland. During the Revolutionary War, Archer was volunteer aide-de-camp to General Anthony Wayne at Stony Point. On June 1, 1779, Archer was made a captain and subsequently a major in the Continental Army.
Archer was elected as a Democratic-Republican to the Seventh, Eighth, and Ninth Congresses, serving from March 4, 1801 until March 3, 1807. In 1810, Archer documented a case of superfecundation, more specifically called "heteropaternal superfecundation," in which a Caucasian woman gave birth to mixed twins — one Caucasian, one Afro-Caucasian - after having had intercourse with two men of differing race within a few weeks.
He was a member of the Revolutionary committee from 1774 to 1776, and later raised a military company during the American Revolutionary War. He was a member of the first state constitutional convention of 1776, and served in the Maryland House of Delegates from 1777 to 1779.
Married Catherine Harris, 10 children (5 became doctors), including Stevenson.