The first part of the word is generally taken to be identical with the verb to jaunt, now only used in the sense of to go on a short pleasure excursion, but in its earliest uses meaning to make a horse caracole or prance, hence to jolt or bump up and down. It would apparently be a variant of jaunce, of the same meaning, which is supposed to be taken from old French jancer.
It was a popular mode of transportation in 19th-century Dublin popularized by Valentine Vousden in a song by that name. Jaunting cars remain in use for tourists in some parts of the country, notably Killarney where tours of the lakes and national park are popular.
The colloquial name for the driver of a jaunting car was "jarvey", referenced in the song "The Jarvey Was a Leprechaun" by Val Doonican.
When William Wordsworth toured Scotland in 1803 with his sister Dorothy and friend Samuel Taylor Coleridge, Coleridge procured an Irish Jaunting Car for the journey from a friend in Somerset. Managing the horse and car proved difficult for Wordsworth, who had appointed himself as the driver.
Jaunting cars are still used for tourist sight-seeing trips, particularly in Killarney in County Kerry.