The finger presses the plastic button. The little black disc drops down and the needle finds the groove. Within seconds the younger set of Mad Men characters, gathered after work in their three-button suits and twinsets to celebrate the acceptance of Peggy Olson's first stab at copywriting by her bosses at Sterling Cooper, are dancing on the floor of PJ Clarke's Third Avenue bar. Each dancer is facing a partner, moving on the spot, without touching. Their arms are going from side to side in time with the rhythm while their knees turn in the opposite direction, a contra-rotational movement involving a demure swivel of the pelvis. Their fresh faces are registering surprise and delight. This is something new. The voice coming from the jukebox is that of Chubby Checker, and the song is the Twist.

It's 1961, and a new era is dawning. On the dancefloor, the Twist – which had enjoyed a brief vogue among America's teens the previous summer – is enjoying a second coming that will sweep the world and change the way people move. No more waltz, quickstep or foxtrot. No more rumba or beguine, cha-cha or tango. No more jitterbug or jive. The Twist is a dance for different times.

The absence of body contact is significant. Rather than going through a set of predetermined steps, you are free to use dancing as a means of self-expression, of doing your own thing, though that phrase will not come into use until the 60s have become fully swinging. It is a narcissistic dance, but it also gives you the chance to watch your partner's moves, and read their intentions. And since you are not physically attached to your partner, there is nothing to stop you drifting away to dance with someone else who has caught your eye (of course, you can also have that humiliation visited upon you, and find yourself dancing alone). Finally, there is no leader: here is the first dance in which the genders are created equal.

Fifty years ago, this felt like a revolution. One evening in the late summer of 1961, I was invited to a teenage party at which a very pretty girl and I were the only ones who knew how to do the Twist. The others gathered round, watching eagerly and then trying for themselves this move that seemed to demolish not just the dance styles but the moral and social structures of the past. Their expressions were those of Mad Men's junior account executives and secretaries. For a while, the Twist was ours and ours alone. But not for long.

No one knows how the Twist began. The word was used in connection with dancing in a number of songs during the first half of the 20th century, but the song itself seems to have been written, in its first form, in 1957 by Brother Joseph Wallace of the Sensational Nightingales, a prominent gospel group. Its profoundly secular nature prevented him performing it himself, but when the Nightingales found themselves sharing a Florida hotel with the popular (and very secular) Hank Ballard and the Midnighters, Wallace offered them his song. Ballard modified the melody and chords to fit the conventional 12-bar blues structure, and the following year, at the Cincinnati studio of King Records, the group committed the result to tape.

King Records' boss was Syd Nathan, an archetypal cigar-chomping philistine who liked the Twist so little that it first saw the light of day, in January 1959, as the B-side of a ballad called Teardrops On Your Letter, which edged its way into the Top 100. As occasionally happened, however, disc jockeys decided they preferred the Twist and played it at teenage functions. It won particular favour in Baltimore, where the audience at a TV show for local teens made up a dance to go with it.

This was quickly spotted by Philadelphia-based disc jockey Dick Clark, who already had a national following for his own TV show, American Bandstand, which was broadcast in an early evening slot on weekdays. Barely into his 30s, the clean-cut Clark was building an empire, and his cultural impact and commercial power were already enormous. As the author of a book called Your Happiest Years, which included chapters entitled "Good Manners are Good Sense" and "Teenagers and Parents Can Be Friends", he had initial reservations about the propriety of the Twist's pelvic movements. But his sharp business instincts had led him to cultivate mutually profitable relationships inside the music business, and he was quick to suggest to his old friend Bernie Lowe, co-owner of Philadelphia's Cameo and Parkway labels, that it would be a good idea to record a cover version of this new song, to feature on his show.

Lowe selected one of his contracted artists, a cheerful, good-looking, puppy-fattish 19-year-old called Ernest Evans whose professional name, Chubby Checker, paid homage to Fats Domino, but whose previous recordings had provoked little response. The job of making the record was given to Dave Appell, a former dance-band guitarist who had become Cameo-Parkway's house arranger. Appell changed the underlying rhythm from the jazzy shuffle of Ballard's original to an even eight-to-the-bar feel derived from Latin music – a pattern that became identified as the Twist beat – and inserted a gritty tenor saxophone solo.

Chubby Checker and the Twist were duly given their first national exposure in August 1960 to the broader audience offered by Clark's Saturday night show, broadcast from New York. The singer gave the audience his famous advice on how to master the dance – "Just pretend you're wiping your bottom with a towel as you get out of the shower, while putting out a cigarette with both feet" – and a month later the record was No 1 in the national charts. By the end of the year, however, it had been forgotten. The Twist already seemed to have gone the way of its predecessors, in double-quick time.

As things turned out, it was only sleeping, and 1961 was destined to be the year of the Twist. At the start of the year Checker had a second No 1 with another dance-craze song, Pony Time, but his next record was a flop. He needed another hit, and in May he was back in Cameo-Parkway's studio to record a song called Let's Twist Again, composed by Dave Appell with Kal Mann, Bernie Lowe's business partner and a former comedy writer. According to Mann, it took all of five minutes to assemble a song clearly designed to do little more than squeeze out the last drop of juice from the original idea. More tuneful than the original, it reached the US Top 10. Elsewhere, meanwhile, events were conspiring to revive what had begun to look like a time-expired fad.

On West 45th Street in midtown Manhattan, a small nightclub called the Peppermint Lounge was setting aside its past as a sleazy leather bar. Owned by the Genovese crime family, and operated by one of its underbosses, Matty "The Horse" Ianniello, as part of a string of strip clubs and gay bars, it had acquired as resident band a young New Jersey group called Joey Dee and the Starliters. They were joined on stage by three teenage girls from Spanish Harlem who had turned up at the club one night in high bouffant hairdos, lavish mascara and matching frocks, with Kleenex stuffed in their bras, and were given a job as the world's first go-go dancers. Later they would become known as the Ronettes, but for now they merely gyrated while the Starliters' high-energy versions of current hits pulled in crowds of young dancers.

Since the club was licensed to hold no more than 178 people, those crowds could never be huge. What counted was not the size but the nature of the audience, for in the late summer of 1961 the Peppermint Lounge enlisted the services of Earl Blackwell, publisher of the Celebrity Register, to arrange visits by a couple of New York columnists: Igor Cassini, who contributed gossip items to the daily Journal-American under the byline "Cholly Knickerbocker", and Eugenia Sheppard, a fashion writer for the Herald Tribune. Both gave prominent mentions to the club, and to the sudden blossoming of the Twist fad, and the warmth of their approval encouraged a flock of celebrities to follow in their footsteps. Before long columnists were recording the presence of Marilyn Monroe, Truman Capote, Greta Garbo, Tennessee Williams, Elsa Maxwell, Noël Coward, Norman Mailer and the offspring of various European royal families. Doormen were being bribed to secure admission for celebrities emerging from Rolls-Royces. As the Cotton Club was to the 20s and Studio 54 would be to the 70s, so the Peppermint Lounge was to its brief era.

Suddenly the Twist was reborn, with a vigour that grew exponentially. Within weeks, Joey Dee and the Starliters were not only topping the charts with Peppermint Twist but entertaining Manhattan's social elite at a charity ball in the Plaza hotel and a party at the Museum of Modern Art. Jackie Kennedy, the epitome of the new carefree spirit of the post-Eisenhower era, did the Twist in a Capri nightspot. Anthropologists and psychologists were asked for their opinions, and the dance made the cover of Time magazine. Checker's The Twist topped the chart for a second time. Arthur Murray, the dance teacher, added it to his curriculum, setting an example followed with some reluctance by Fred Astaire's nationwide chain of academies. Suddenly almost every new record seemed to have Twist in the title, from future classics such as Sam Cooke's Twistin' The Night Away, Gary US Bonds' Twist, Twist Senora and the Isley Brothers' Twist And Shout, to countless examples of exploitative dross. And, inevitably, Hollywood started taking an interest.

Twist Around The Clock was launched on 30 December 1961, with the craze at its height. A barely disguised low-budget rewrite of Rock Around The Clock, it was advertised with the slogan "It's twist-errific!" and featured Checker, Dion DiMucci and the Marcels with such songs as Twist Along, Twistin' USA and The Twist Is Here To Stay. The following day a competing film, Hey, Let's Twist, hit the cinemas, centring on the Peppermint Lounge, with Joey Dee featured as an ambitious young singer and Joe Pesci, a sometime guitarist with the Starliters, making his uncredited screen debut as a dancer in the club scenes (the Ronettes were to have been given roles as the Starliters' girlfriends, until the film's producers clocked their skin pigment). In Britain, where Let's Twist Again reached No 2 in the charts, the film industry followed suit, recruiting Checker to sing something called The Lose Your Inhibitions Twist in Dick Lester's It's Trad, Dad!, which also featured Acker Bilk, Kenny Ball and Chris Barber.

Checker rerecorded Let's Twist Again in German, for an audience also dancing to Der Liszt Twist. In France there was Twist à Saint-Tropez, by Dick Rivers et Les Chats Sauvages, while Johnny Hallyday invited his followers to Viens Danser Le Twist. But by the time the Beatles brought their first album to a climax with their raucous, Hamburg-honed version of Twist And Shout at the beginning of 1963, the whole business had become the inevitable victim of overexposure. To its early adopters, it was history.

The Peppermint Lounge would lose first its celebrity clientele and then its liquor licence, but it could be credited with popularising the idea of the discotheque: a phenomenon that, with or without go-go girls, would be a great deal harder to eradicate than the dance that made its name. What the Twist had done, however, was create a powerful hunger among modernist youth for new dance crazes based on the template of dancing on the spot, with no contact. And so along came the Locomotion, the Fly, the Madison, the Hitch Hike, the Watusi, the Hully Gully, the Frug, the Stroll, the Monkey, the Dog, the Mashed Potato and countless others, including that nameless creation, beloved of mods, in which all movement was reduced to the merest twitch of one knee and a barely perceptible shrug of the shoulders: it was, as New Orleans R&B singer Chris Kenner would proclaim in 1962, the Land Of 1,000 Dances indeed.

And when I watch that Mad Men scene now, there's something in it I recognise from the teenage party of half a century ago: the faces, with their look of joy and discovery. It was only a dance, for heaven's sake, but it opened up a world.