The rule of thirds is a method of creating more dynamic visual compositions. Imagine lines that split the screen into equal thirds both horizontally and vertically, creating nine equal parts. Main points of interest are placed on or near any one of these lines. In some of the examples above by Vermeer, Caravaggio, Rembrandt and Fragonard, key points of interest are placed where horizontal and vertical lines intersect, sometimes called "power points".
There are many other concepts that can lead to strong compositions, but this is one the simplest concepts to begin using