It seemed like a good idea at the time. What New York clearly needed was a Society of Independent Artists. It would hold an annual exhibition, to which any painter or sculptor could submit work, and somehow it would be displayed. There wouldn't be any jury or judges; the artist would simply have to pay a fee of $6, and their work would be displayed in the exhibition hall of the society.

Except, that is, for one unforeseen development. One day in spring 1917, the organisers of the forthcoming exhibition stood and gazed in consternation at one of the submissions. The sculpture – is that what it was? – was accompanied by a letter from one Mr Richard Mutt, who enclosed his six dollars. The object bore the signature of the artist – R Mutt – and it had a title, Fountain. But the "artist" obviously hadn't sculpted the object himself. It was an industrially produced object. Not a fountain, but, well, a urinal.

In fact, it was a urinal of the most common variety, in white porcelain, lying on its back. You can see it, from tomorrow, at Tate Modern's exhibition Duchamp, Man Ray, Picabia – and it's fair to say that the response to it in 1917 in New York may well be replicated by some visitors, in 2008, in London.

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Ninety-one years ago some discussion took place and, by a small majority, it was decided that the object should be returned to Mr Mutt with the comment that whatever incarnation his object was, it had no darn place in an exhibition of art. In a very short period, it then emerged that there was no Richard Mutt. The man behind the bizarre and incredible submission proved to be one of the directors of the society itself, an amusing and audacious French artist, recently arrived in America, called Marcel Duchamp. He resigned from the society; and art was never the same again.

Were they complete fools to turn down a great classic of modern art? In fairness, the directors could never (in 1917) have been asked to pass judgement on anything that remotely resembled Fountain. While almost every artist since the beginning of time has adapted, refined and developed art as it was handed down to him, it is not given to very many artists, without resource to the changing means of technology, to invent an entirely new art. In fact, it might be that, in the long centuries since the invention of easel painting, only Duchamp has carried out the sort of revolution represented by that urinal.

Such is the importance of Fountain that, in December 2004, it was voted the most influential work of modern art by 500 art-world professionals. In contemplation, that seems something of an understatement: with this single "readymade" work, Duchamp invented conceptual art and severed for ever the traditional link between the artist's labour and the merit of the work.

It couldn't have happened anywhere but America. Duchamp, at the height of the furore, declared that "the only works of art America has given are her plumbing and her bridges". America, too, was the land of the huckster and the charlatan – one of the meanings of Duchamp's pseudonym refers to the gambler Mutt in Bud Fisher's famous Mutt and Jeff cartoon strip. It would be wrong to neglect the element of the huge practical joke in Fountain and its general air of disconcertingly ridiculing the art world as well as, with complete seriousness, reconfiguring everything it thought it believed about art.

The objections to Fountain began immediately, and have never gone away. Duchamp, in a brief response in the avant-garde journal The Blind Man in 1917, summarised them: "Some contended it was immoral, vulgar. Others, it was plagiarism, a plain piece of plumbing." These objections seek to pin down the object and its meanings to the purely functional. There can be no meaning beyond that, and no artistic merit.

But it's quite extraordinary how, once the urinal is removed from its customary setting, laid on its back like a turtle and given a signature, its meanings immediately multiply. For some reason, it hasn't been much remarked on that the object must have been an unfamiliar and exotic sight for half the audience.

Although Duchamp defends it against indecency by saying that it is a common sight in plumbers' show windows, it would only be really familiar to his male audience. Public lavatories were, indeed, a very recent development in the modern world, and there was little provision for women at all. It will be remembered that the "furmity woman" in Hardy's The Mayor of Casterbridge (1886) is arrested for "committing a public nuisance" in a large market town. There were, simply, no private alternatives. So what did women do? We have the witness of the English sexologist Havelock Ellis that in the 19th-century fashions of crinolines, ladies simply paused, ideally over a gravel path, and micturated while gazing into the middle distance. The display of a urinal, then, was not commonplace at all, but extraordinarily modern, and one designed to be highly exotic and unfamiliar to half the audience.

Duchamp had started thinking of the idea of the "readymade" a year or two earlier. His first was a mounted bicycle-wheel on a stool, which he said he simply liked looking at. But many of the others contain some element of the exotic alongside their commonplace, manufactured element. A snow shovel enchanted Duchamp, who had never seen such a thing until he came to America from France. A bottle-dryer, conversely, must have been an unfamiliar and peculiar sight for his American circle.

Fountain divides its audience immediately along the lines of sex, who will in turn find it banal and familiar, or exotic and forbidden. Sexual attraction and sexual difference was one of the most enduring obsessions of Duchamp, and often the peculiar conventions of the erotic furnish him with his most dazzling inventions. His early painting Nude Descending a Staircase gets all its outrageous fun from the convention that a nude is decent and respectably aesthetic if seated or standing. What nudes can't do without vulgarity is walk down a staircase. They would pretty soon be walking from the landing to the hallway, from the drawing room to the kitchen; and there is nothing more convincingly obscene than a cold burst of the everyday, as the "readymades" extensively show.

The work known as The Large Glass, or The Bride Stripped Bare by Her Bachelors, Even, made between 1915 and 1923, similarly divides according to sexual difference. The bride, in the upper panel, is all fluidity and biomorphous shapes. The bachelors inhabit a world of arid, mechanical self-gratification, including sexual imagery drawn from an earlier Duchamp painting of a chocolate grinder.

And his last great work, Etant Donnés, an extraordinary peep-show tableau in Philadelphia, throws the viewer back on his own voyeurism in a mysterious display of withdrawn pornographic femininity. Duchamp liked to move fluidly from one sex's viewpoint to another, creating some of his best work in the persona of Rrose Sélavy (there is a hilarious photograph of Duchamp in Rrose drag by Man Ray). But he never confused them with each other.

Fountain, in a way, condenses all those sexual meanings with great concision. It is a masculine property, but clearly one which has the feminine properties of receiving a man's fluids. That doesn't sound much like a fountain, however; and one rather comes to the conclusion that the source of the fountain's jets, the bouche as it is called in French, are the parts of a speculative masculine observer. Just as the aesthetics are solidly based in, and rather rely on, the viewer's reactions, so its meaning depends on his reconstructing the object's use in his head. Anyone who thought it obscene would have to supply quite a train of thought to the object's faux-innocence.

Not everyone has been satisfied to think only of the object's use. In recent years, one "artist" has succeeded in urinating into Fountain, and more have claimed to have done so. Some of Duchamp's works positively invite ad hoc collaboration with the eager, playful observer, such as the toybox versions of his greatest hits made to fit in a single suitcase. One early admirer picked up the curiously heavy cage filled with marble sugar-cubes entitled Why Not Sneeze Rrose Sélavy and was told by a bored guard: "Don't you know, you ain't supposed to touch that crap?"

But the great Fountain doesn't invite intervention. We are just supposed to look at it and marvel. It is, oddly enough, an extraordinarily beautiful object. Its lovely curves have the warmly shifting shape of many of Duchamp's renderings of femininity. In a famous photograph of 1917 by Alfred Stieglitz, Fountain has reminded many people of a grossly oversimplified erect penis and testicles. In other representations, under bright light and with its shadows clearly outlined, it has been likened to the shape of a modest woman with head covered.

It is difficult to understand why those Americans who delight in discovering likenesses of the Virgin Mary in the most unlikely objects have never turned their attention to Duchamp's Fountain. In many photographs, that is exactly what it looks like, and I guess Duchamp turns it on its back to exacerbate this iconic, blasphemous resemblance.

Fountain, as full of meaning as an egg is full of meat, changed art for ever. It had always been clear to thoughtful observers that the link between an artist's skill and the merit of his work was a false one. Some of the greatest painters in the world, such as Watteau or Goya, possess a limited technique, and many of the most brilliantly virtuosic and intricate produce art of no ultimate value.

There has never been any value in the proposal that the harder an artist works, or the more skilfully detailed his craft, the better the work of art in the end. The link between labour and product was not decisively broken until Duchamp, however.

Perhaps the larger context helps us to understand why this happened in 1917, and not before. Tristan Tzara, the founder of Dada and a thinker in tune with Duchamp, said in his 1918 "Dada Manifesto" that "a work of art should not be beautiful in itself, for beauty is dead. A work of art is never beautiful by decree, objectively and for all. After the carnage we still retain the hope of a purified mankind."

Duchamp came from a small town near Rouen, very close to the battlefields of the Great War. The war was a massive discrediting of the powers of authority, and what men like Tzara and Duchamp were doing was dismantling the shaman-like powers of the virtuoso artist, the powers of judgement of critics and the juries that turned out to rule even the New York Society of Independent Artists. As the century went on, the power and enchantment of the "readymade", in the hands of men like Joseph Beuys, would turn out to be precisely its democratic, unambitious scale.

Many German artists, after the Second World War in particular, were uncomfortable with virtuoso artists commanding a mesmerised following. Beuys's response, always, to the philistine response of "I could do that – I could find something and put it in an art gallery; anyone could" was always: "Of course you could; why on earth do you believe the creation of art to be something beyond your capacities? Who taught you that?"

The liberation of art within Duchamp's Fountain and the other "readymades" took a long time to appreciate. The version displayed at Tate Modern from this week is one of half a dozen replicas certified by Duchamp, 50 years after the original was first exhibited, and much of Duchamp has had to be reconstructed in this way. Most of the original "readymades" were thrown away or misplaced, or perhaps, as in the case of the hatrack, it just started being used for its original purpose.

Duchamp was a paradoxical master; somebody who made both a complex theoretical contribution, and the most lovable and forgivable of charlatans – anyone who thinks there is no element of the charlatan in Duchamp has understood, frankly, nothing. He was somebody who opened the way for the slacker, the anti-authoritarian, the Zen contemplator of emptiness and small moments of anonymous poetry, while remaining the most charismatic, disciplined and powerfully individual of artists.

Some people will always maintain that Duchamp's gesture is not art, perhaps mistaking a statement of factual category for a critical judgement of excellence. Others go on maintaining that it only needed to be done once, and nothing afterwards would ever have the shock or the fascination of that initial display of 1917.

That misses the point. Duchamp knew perfectly well that, with the public display of Fountain, he was taking off a lid that was screwed down tightly, with immediate and explosive effect. There was never going to be the slightest possibility that it could be screwed down again afterwards. Art had escaped, and was enveloping the world.

Duchamp, Man Ray, Picabia is at Tate Modern, London SE1 (020-7887 8888; www.tate.org.uk) from tomorrow to 26 May

Fountainhead: Duchamp’s disciples

By Rob Sharp

Richard Hamilton

Hamilton, 86, has been described as "era-defining" and "Pop Art's intellectual father". He is one of Britain's most distinguished Pop artists; a "Pop Daddy", if you will. He designed the cover of The Beatles' White Album and the poster in 1968. Hamilton has written extensively about Duchamp. He says: "Duchamp was iconoclastic. He questioned anything that happened before, and he clearly tried to create a work which was unique. I say a work; I mean, his life work; his life's work was something that was completely unique and had never been done before and I think that's quite deliberate."

Cai and Xi

The Japanese performance artists achieved fame by romping on Tracey Emin's My Bed at the Tate Gallery in 1999. She was not amused; they said they were searching for a "new route" in art. A year later, the duo urinated on Duchamp's Fountain at the newly opened Tate Modern, but were prevented from soiling the work directly by a Perspex case. Tate said they were threatening "works of art and our staff" and banned them from the premises. Asked why they felt they had to "add" to Duchamp's work, Cai said: "The urinal is there – it's an invitation. As Duchamp said himself, it's the artist's choice. He chooses what is art. We just added to it."

Grayson Perry

The Turner Prize-winning transvestite potter Grayson Perry admits his work is influenced by Duchamp, but is keen to distance himself from him in the cross-dressing department. Earlier this month, Perry, 47, said that unlike Duchamp, he doesn't have a female alter ego. For those who weren't quite sure what he meant, he explained: "I'm a transvestite – a man in a dress."

Perry's views on Duchamp's contribution to art are more profound. "Since Duchamp a century ago, anything an artist chooses can be art. Now, when judging it, I have to make sure that I am using the specific criteria that measure quality in an art work. I look at a work now and think: I like this. But am I drawn to it because the juicy red bits would look great next to my yellow sofa or because I agree with its political statement, or because it is a masterpiece?"

Richard Wentworth

Wentworth emerged during the 1980s as one of Britain's most striking sculptors. He makes direct reference to the subversive Gallic urinal merchant in a recent piece called If Only, in which Duchamp's 1914 Bottle Rack (in France, these were used to dry empty wine-bottles) is spliced by a vertical piece of glass. The 61-year-old said recently: "I think Duchamp is like accepting your grand parentage. It is like [such artists] made an enormous space in which you were allowed to be 50, 60 years later."

Peter Blake

The pointy-bearded Pop artist once depicted Duchamp travelling around the world on a "rock'*'roll tour bus" in a famous collection of paintings. In this series, which were featured in a retrospective last year, the artist meets the Spice Girls and Elvis Presley, and even plays chess with Tracey Emin. "Duchamp was very important to me as a young artist," Blake, 76, said recently. "He's also been very important to the YBAs [Young British Artists]. They couldn't have existed without him. Which is why I put Tracey Emin in. So in a way I'm thanking him by sending him on this wonderful world tour in the afterlife." Blake has also said: "Thank you to Duchamp. I am giving him the gift of living forever. Everybody in the 21st century owes a debt to him."

Damien Hirst

The world's most lucrative contemporary artist has paid homage to all and sundry in his various splutterings about contemporary art – but Duchamp was responsible for launching the movement on which his considerable fame and fortune were originally founded. In one recent interview, Hirst, 42, declared: "Duchamp and Picasso I'm interested in... What about [Richard] Hamilton? Don't you think he should be important? It just seems a little weird to me that [many artists' influences are] all Americans. The big ones. Throughout history... It seems a bit of an imbalance, d'you know what I mean?"