The Separation of Layers within an HTML document is a somewhat recent convention within the Web design community. It is centered on standards for the ultimate purposes, shared with all other best practices, of cross-browser compatibility and a concept of the ideal functionality for the Web. The focus is more on the latter for this specific convention, but cross-browser compatibility augments how that goal is achieved, to the point of making exceptions where necessary.
The specific ideal behind it is the separation of concerns via modular programming, i.e. the maximal independence of each piece of software, or module, from each other piece. A module, in any sub-field of programming, typically follows the Unix philosophy to "do one thing well" and is therefore as small as it can be while still doing its job well. ("Programming" here is used loosely, although the use of "software" for HTML and CSS is technically accurate.)
When the concerns for a page's structure and the content's semantics are separated, they are commonly divided at the border between block-level and inline elements. While the template provided by the server may contain inline elements, they are usually not within the area designated for the separated content. Additionally, most block-level elements can validly contain all the others, so the content may add to the structure.
Semantics are specified with HTML elements such as tables, forms, lists, headings, paragraphs, and generally any that is inline. They affect how a typical Web browser or other program, such as a screen reader or Web crawler, will "understand" the content. Some are rendered in typical browsers in a similar manner as those used for formatting, although their purpose is significantly more specific.
Wikis such as this one provide syntax which is primarily geared toward such markup, although the div element is provided for structural purposes (perhaps the only such exception). Some unconventional, even non-standard, tags and attributes are allowed but the same best practices apply to wiki authoring as to Web design.
Style affects the color, size, decoration, and, with some constraints applied by the HTML structure, the placement of elements. Handles for CSS selectors, such as class and id attributes, are typically the only sign within the HTML that CSS is being used, allowing stylesheets to be interchanged seamlessly in true modular fashion.
This and other MediaWiki wikis are a great example of this, even going to the extent of allowing the user to specify CSS styles in addition to those supplied by the wiki's staff and the wiki software itself.