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.