as a term actually originated in the USA, in the early-mid 1920s, and originally referred to jug band
music, simple blues
-related music that was often played at house parties, often on improvised instruments. The term fell into disuse when the style of music itself did.
However, in the 1950s there was the curious trad jazz boom in Britain, where for some inexplicable reason the most popular music among teenagers became bands like those of Humphrey Lyttelton, Acker Bilk, George Melly and Chris Barber, who based their entire careers on trying to emulate as precisely as possible the music of 1920s Dixieland musicians like Louis Armstrong, Sidney Bechet and Bix Beiderbecke.
This craze, like most post-swing, pre-rock musical fads, didn't last long with the public, but did spark a second craze, with possibly more impact. Chris Barber's band was more adventurous than many of the other bands, and would often back visiting US bluesmen such as Muddy Waters (this continued even into the 80s - there is a famous video of them backing Dr John at the Marquee club). In between sets various members of the rhythm section of the band would often play impromptu blues jams, two of these members (Cecil Davies and Alexis Korner) going on to found R&B bands that would nurture the talents of, among many others, the young Rolling Stones.
However, guitarist Anthony Donegan was less enamoured of the current electric blues, and preferred acoustic folk blues artists such as Leadbelly and especially Lonnie Johnson, from whom he took his stage name, becoming 'Lonnie Donegan'. One of the songs he busked between sets was Leadbelly's classic Rock Island Line, which released as a single in 1956 by 'Lonnie Donegan's Skiffle Band' became a surprise huge hit in the UK and revived the style.
In truth, the skiffle revival lasted only a few months - none of the new practicioners had Donegan's talent, and Donegan himself soon turned to recording novelty songs such as Does Your Chewing Gum Lose Its Flavour On The Bed-Post Overnight after he stopped having hits with the Leadbelly and Woody Guthrie songs that he initially gained popularity with. But its impact was important, because like punk twenty years later, skiffle was 'do-it-yourself' music, made on improvised instruments.
Many, many young bands gained their first experience as skiffle bands - as noted in the two writeups above, the most notable being the Quarrymen (who at the time they were a skiffle band actually consisted of John Lennon, Rod Davis, Colin Hanton, Eric Griffiths and Len Garry) - but almost all quickly changed to playing rock & roll as their finances improved and they could afford real basses rather than tea-chests and real percussion rather than washboards.
There are still a few people who play skiffle music (Donegan made occasional returns to the form until his death), but it has degenerated into a slavish recreation of old, dead musical forms. The most obvious reason for this is that the society in which tea-chests and washboards were common household items that could be recycled for use as musical instruments simply doesn't exist any more. The only time I've ever seen a washboard in my life was in a music shop...