“Stile and rail” is a type of wood construction for doors and other panel-like objects. The horizontal piece is called the rail, the vertical piece is called the stile.
In ye olden days, before the advent of plywood and other engineered panel products, doors were made from solid planks of wood. If you have a wood floor, you may know that gaps appear and disappear throughout the year depending on the season, humidity, species of wood, etc. For a door or any large panel, this kind of expansion and contraction wreaks havoc on the fit of the door creating gaps and binding. To overcome this problem, stile and rail construction became the preferred solution for quality carpentry. The boards that make up the panel, the center part, are slotted into the surrounding stile and rail and can slightly move, expanding and contracting, within this frame without effecting the overall dimensions and stability of the frame. The familiar look of stile and rail, the surrounding “frame” and inset panel, was not simply an aesthetic choice, but the result of a technological solution.
For the last almost 100 years, typical doors and cabinets are made with a variety of materials, often composite wood products, that are dimensionally stable without the need of the use of stile and rail technology. However, the familiar look of stile and rail has persisted, especially in door styles.
So, remember that “stile” is not “style”, but “stile and rail” has become a style. Simple.