Archer, the son of a Warwickshire squire, was educated at Trinity College, Oxford, and then spent four years abroad. After receiving several lucrative royal appointments, he bought the manor of Hale, Hampshire, in 1715 and rebuilt the house and church.

Archer's dynamic work borrowed much from the 17th-century Italian architects Gian Lorenzo Bernini (Bernini, Gian Lorenzo) and Francesco Borromini (Borromini, Francesco). Although he was not as original as some of the prominent English architects of his time, he was considered important. Most of his designs were executed from 1705 to 1715, including the north front of Chatsworth House (c. 1705) in Derbyshire, Heythrop House (1707–10) in Oxfordshire, a garden pavilion at Wrest Park (1709–11) in Bedfordshire, Roehampton House (c. 1712) in Surrey (now in Wandsworth, London), and the churches of St. Philip (c. 1710–15) in Birmingham, St. Paul (1712–30) in Deptford, and St. John (1713–28) in Westminster. The last two resulted from his appointment in 1711 as a commissioner for the building of 50 new churches.