means A device that directs data packets between networks, deciding where your internet traffic goes next on its way to and from your devices.

from From the verb 'route,' meaning to send along a course, which comes via French 'route' from Latin 'rupta (via)' — literally a 'broken way,' a path beaten or cut through the land. The '-er' simply marks the thing that does the routing. (Note: in woodworking, a 'router' is the spinning tool that cuts grooves — same word, since it too 'routs out' a path; in networking the metaphor is purely about steering traffic down the right road.)