A division is a structured component of a book or article that would be recognized by most any reader. They are essential to the organization of a PreTeXt project. Notice that we use the generic term division, since a <section> is just one example of a division.
divisions are <book>, <article>, <part>, <chapter>, <section>, <subsection>, <subsubsection>, and <paragraphs>. Their use is fairly intuitive, though there are some restrictions, so please read on.
A <book> must contain at least one <chapter>, which may contain <section>, <subsection>, and <subsubsection>. The tag <part> is planned for books, between <book> and <chapter>. It will function in two user-selectable ways: structural (e.g. numbering will reset), or decorative (merely inserting a decorative page between two chapters and sectioning the Table of Contents).
An <article> is simpler and shorter than a book. It might be really simple and have no divisions at all, or it may have <section>s. It cannot have <chapter>s, as that would be a <book>. Within a <section>, <subsection>s and <subsubsection>s may follow.
Divisions must nest properly and may not be skipped. So a <section> cannot contain a <chapter> and a <subsection> may not be contained in a <chapter> without an intervening <section>.
A division must contain a <title>, and may contain one or more index entries (see <<topic-index>>), which should appear before anything else. Any division may be unstructured, with just a sequence of top-level content such as paragraphs, figures, lists, theorems, etc. Or a division may be structured, and in this case it must follow a prescribed pattern. There may be a single, optional <introduction>, filled with top-level content, followed by a sequence of at least one of the appropriate divisions, ending with a single, optional <conclusion>, filled with top-level content. It is an error to begin with a run of top-level content inside a division and then begin to use divisions. (The solution is to make the initial content an <introduction> and/or one or several divisions.)
There are exceptions to the above. For one, <paragraphs> is an anomalous division, as a sort of lightweight sectioning command. It may appear in any division, at any location within a division, it may not be divided further (it is a leaf of the document tree), it never gets a number, and its title is formatted in a subsidiary way. I especially like to use this in a two- or three-page <article> that has no other divisions at all. Typical presentation has the title in bold, without much change in font size (if at all), inline with the first paragraph, and perhaps a bit of vertical space as it begins and ends. Despite the name, it may contain more than just paragaphs, so may contain any top-level-content that would go in any other division.
Two other anomalous divisions are <exercises> and <references>. These can function as a further division of any other division. So for example, an <exercises> could be a peer of several <section>s, contained within a structured <chapter>, and in this case would behave like the other <section>s with regard to numbering and presentation. Detail on allowed content and its behavior are in Section 6.1 and Section 6.2, respectively.