A layout engine, or rendering engine, is software that takes marked up content (such as HTML, XML, image files, etc.) and formatting information (such as CSS, XSL, etc.) and displays the formatted content on the screen.
It "paints" on the content area of a window, which is displayed on a monitor or a printer.
A rendering engine is typically used for web browsers, e-mail clients, or other applications that require the displaying (and editing) of web contents.