What is the correct name for aside boxes that text has to flow around?
For an example of what I mean, please look at this article. I'm talking about boxes like the one that starts with "Sign Up for the Opinion Today Newsletter".
A pull quote (also known as a lift-out pull quote) is a key phrase, quotation, or excerpt that has been pulled from an article and used as a graphic element, serving to entice readers into the article or to highlight a key topic. It is typically placed in a larger or distinctive typeface and on the same page. Pull quotes are often used in magazine and newspaper articles, annual reports, and brochures, as well as on the web. They can add visual interest to text-heavy pages with few images or illustrations.
In publishing, a call-out or callout is a short string of text connected by a line, arrow, or similar graphic to a feature of an illustration or technical drawing, and giving information about that feature. The term is also used to describe a short piece of text set in larger type than the rest of the page and intended to attract attention.
However in your example, it's merely an advertisement, unrelated to the rest of the page content.
It's a sidebar.
a short article in a newspaper or magazine, typically boxed, placed alongside a main article, and containing additional or explanatory material.