Sam Phillips, soon gave a copy of Elvis' record to local disc jockeys Dewey Phillips (no relation) of WHBQ, and on July 7, 1954, Dewey Phillips played "That's All Right" on his popular radio show "Red, Hot & Blue". Interest in the record was so intense that Dewey reportedly played the record 14 times and received over 40 telephone calls. Girls especially went wild over Elvis' song!
Elvis' recording of "That's All Right" was officially released on July 19th 1954 with "Blue Moon of Kentucky" as the B-side, and sold around 20,000 copies. It is #112 on the 2004 Rolling Stone magazine list of the "500 Greatest Songs of All Time"
The sequence of events that led to this moment began when a young truck driver walked into the offices of Sun Records in the summer of 1953 and paid $3.98 plus tax to make an acetate record as a birthday present to his mother. Elvis recorded "My Happiness" and "That's When Your Heartaches Begin," and his business partner Marion Kreisler heard something that made her write down "Good ballad singer. Hold" in her notes. In early July 1954, Phillips finally sent two of his favorite session musicians, guitarist Scotty Moore and bassist Bill Black, to go meet with Elvis. After talking and jamming a bit with Presley, Phillips decided to schedule a recording session with Presley for July 5th 1954.
The first several hours of the July 5th session did nothing to convince Sam Phillips that Elvis was the one he'd been looking for. After numerous takes and re-takes of songs, Phillips called for a break. Rather than shoot the breeze with his fellow musicians or step outside for a breath of fresh air, Elvis began to mess around on the guitar, playing and singing "That's All Right," but at least twice as fast as the original. Through an open door in the control room, Sam Phillips heard this unfamiliar rendition of a familiar blues number and knew he'd found the sound he'd been looking for. "[Phillips] stuck his head out and said 'What are you doing?'" Scotty Moore later recalled. "And we said, 'We don't know.' 'Well, back up,' Sam said, 'try to find a place to start, and do it again.'"
Phillips continued recording with Elvis over the next two evenings. Released to Memphis radio station WHBQ just two days after it was recorded, and then as a single two weeks later, Elvis Presley's "That's All Right (Mama)" became an instant regional hit and set him on his path toward worldwide stardom.
sources: history.com and wikipedia.com