This pair of marks, whether single (‘ . . .’) or double (“ . . .”) is known as inverted commas in the UK, and in the US, I believe, as quotation marks, although both terms seem to be fairly widespread.
According to David Crystal, writing in ‘Think On My Words: Exploring Shakespeare’s Language’, ‘The mark started to appear in English during the 1590s, but did not come into regular use until the late eighteenth century.' This accords with the earliest citation for inverted comma in the Oxford English Dictionary which is from 1789: ‘Two inverted commas are generally placed at the beginning of a phrase or a passage, which is quoted or transcribed from some author, in his own words.’