Engine names a machine that converts energy into mechanical force and motion — and, more broadly, anything used to effect a purpose, from a railroad locomotive to a piece of software performing a fundamental function inside a larger program.
The oldest English sense has nothing to do with machinery. Around 1300 an engine was skill, craft, innate ability — and equally its darker cousins, cunning, deceit, and trickery. The word came through Old French engin ("skill, wit, cleverness," and also "war machine") from Latin ingenium, "innate qualities, ability, inborn character," built from in- and gignere, "to beget." An engine was etymologically that which is inborn: native cleverness first, then the contrivance that cleverness produced. By Late Latin the word had already narrowed toward siege equipment, and for centuries an engine in English was above all a machine of war. Only in the eighteenth century did the modern sense — a device converting energy into mechanical power — take hold, and through the nineteenth it belonged chiefly to steam.
The root is still visible across the family. Engineer is a fourteenth-century constructor of military engines long before it is anyone who designs bridges, and the civil qualifier exists precisely to mark the departure. Ingenious preserves ingenium nearly intact. Gin, as in cotton gin, is a worn-down engine. The computing sense follows the same logic of a component doing the fundamental work while the rest of the program arranges itself around it — a rendering engine, a game engine, or the search engine at the center of 📝Search Engine Optimization (SEO).