Usually HTML pages can be considered two-dimensional, because text, images and other elements are arranged on the page without overlapping. There is a single rendering flow, and all elements are aware of the space taken by others. The {{cssxref("z-index")}} attribute lets you adjust the order of the layering of objects when rendering content.
In CSS 2.1, each box has a position in three dimensions. In addition to their horizontal and vertical positions, boxes lie along a "z-axis" and are formatted one on top of the other. Z-axis positions are particularly relevant when boxes overlap visually.
(from CSS 2.1 Section 9.9.1 - Layered presentation)
It means that CSS style rules allow you to position boxes on layers in addition to the normal rendering layer (layer 0). The Z position of each layer is expressed as an integer representing the stacking order for rendering. Greater numbers mean closer to the observer. Z position can be controlled with the CSS {{ cssxref("z-index") }} property.
Using z-index appears extremely easy: a single property, assigned a single integer number, with an easy-to-understand behaviour. However, when z-index is applied to complex hierarchies of HTML elements, its behaviour can be hard to understand or even unpredictable. This is due to complex stacking rules. In fact a dedicated section has been reserved in the CSS specification CSS-2.1 Appendix E to explain these rules better.
This article will try to explain those rules, with some simplification and several examples.
Note of the author: Thanks to Wladimir Palant and Rod Whiteley for the review.