The Douay-Rheims is a Catholic translation of the Bible into English, completed in its original form in 1609. It was the work of Catholic scholars who fled from Protestant persecution in England to France, where they founded English-speaking Catholic colleges. The New Testament was translated by the Catholic college at Rheims (nowadays generally referred to as Reims), first published in 1582; while the Old Testament was published in 1609-1610 by the Catholic college at Douay (nowadays spelt Douai), hence the name Douay-Rheims.
However, what is commonly called the Douay-Rheims nowadays is not this original translation completed in 1610, but the revision thereof by Bishop Richard Challoner published in 1749 (with subsequent editions in 1750 and 1752). The original Douay-Rheims, without Challoner's revisions, is not in common usage.
The Douay-Rheims version included the entirety of the Catholic canon, including those books rejected by Protestants. It also included the Clementine Apocrypha, the Prayer of Manasses, 3 Esdras (which the KJV calls 1 Esdras), and 4 Esdras (which the KJV calls 2 Esdras). These three books are not considered by Catholics to properly form part of the Bible, but were deemed worthy of historical preservation.