Carl Wilhelm Scheele, working as an apothecary in Sweden, prepared oxygen at some point in 1771 or 1772, described it as fire air, and did not publish until 1777. Joseph Priestley, in England, obtained it on 1 August 1774 by focusing sunlight onto mercuric oxide, noted that a candle burned in it with remarkable vigour and that a mouse lived unusually long in it, and published in 1775. Antoine Lavoisier, in Paris, heard of Priestley's result from Priestley himself over dinner in October 1774, repeated the experiment, and understood it.
That is the whole dispute, and every part of it is true. Scheele made it first. Priestley published first. Lavoisier explained it, named it, and built a theory of chemistry on it. National accounts have tended to promote whichever of the three suits them, and the honest summary is that the discovery was genuinely divided and that the three contributions were of different kinds.
What Priestley and Scheele had was a gas. What Lavoisier had was combustion. The prevailing account held that burning bodies release a substance called phlogiston; Lavoisier showed that they instead combine with something drawn from the air, and that the products weigh more than the starting materials rather than less. He named the gas oxygene, acid-former, on the mistaken belief that it is present in all acids, and the error is fossilised in the name.
The consequence reaches beyond chemistry. Phlogiston was the last framework in which alchemical practice and academic chemistry still shared a vocabulary. Lavoisier's list of elements in 1789, and his insistence that an element is simply whatever cannot be broken down further by experiment, replaced a system of principles and qualities with a system of substances. Alchemy did not so much lose the argument as lose the language the argument had been conducted in.