Oxygen was "discovered" by a number of scientists over a fairly brief time period. First discovery is normally credited to the Swedish scientist Carl Wilhelm Scheele in 1772. He called his discovery "fire-air" because it supported combustion.
Unfortunately (for him), Scheele's findings weren't published until 1777. In the interim, British clergyman and chemist Joseph Priestley, in 1774, discovered oxygen independently. He called it "dephlogisticated air." (Aren't you glad we don't call it that today?)
Across the English Channel, French scientist Antoine Lavoisier claimed to have discovered oxygen independently in 1775. Priestley, however, claimed that he had visited Lavoisier in 1774 and talked with him about his (Priestley's) experiments, so the independence of Lavoisier's "discovery" is questionable.
Lavoisier did contribute two important things to the discovery of oxygen, though: He was the first to deduce that oxygen was a separate element, and he gave it the name by which we know it today. Oxygen, from Greek roots that mean "acid-producer," was so named because of Lavoisier's belief that oxygen was present in all acids.