[lemoncucumber]

You're thinking of pull quotes, which can be confusingly similar to block quotes but are a different thing:

> It seems that many people (myself included) confuse blockquotes and pull quotes.

> The main purpose of a blockquote is to separate a large section of text — quoted from an outside source — that is relevant to the source material at hand.

> A pull quote is a section of the article pulled out of its context and repeated to give either emphasis, or to aid the reader in scanning the article.[0]

Unfortunately HTML doesn't provide a pull quote element so you have to make do with applying some CSS to a <blockquote>, or <p>, or whatever to get the desired visual effect.

[0]: https://www.studiopress.com/how-to-use-block-quotes/

[derefr]

I think a pull-quote might be one of the intended uses of <aside>.

(Though really, I don’t think HTML “likes“ pull-quotes — they’re a denormalization of the information conveyed by the markup. The ideal from WHATWG’s perspective would likely be some sort of intra-document transclusion reference, such that the pull-quote could “sample” the text out of where it already is on the page, without mirroring it.)