Short answers, pulled from the story.
John Lennon founded the Quarrymen in Liverpool in 1956, forming the initial skiffle group in November of that year with Pete Shotton and Eric Griffiths. The group took its name from a line in the song of their school, Quarry Bank High School.
Paul McCartney met John Lennon on the 6th of July 1957 at the St. Peter's Church Rose Queen garden fete in Woolton, where Ivan Vaughan made the introduction. McCartney made his official debut with the Quarrymen on the 18th of October 1957 at the New Clubmoor Hall in the Norris Green section of Liverpool.
The Quarrymen booked a session on the 12th of July 1958 at Phillips' Sound Recording Services, a studio at 38 Kensington, Liverpool, run by Percy Phillips. They recorded straight to disc using a single microphone, producing "In Spite of All the Danger" and Buddy Holly's "That'll Be the Day". Both recordings were later released on the Beatles' album Anthology 1.
By March 1960, Lennon and Stuart Sutcliffe settled on the name the Beatles, chosen for its double meaning and as a reference to Buddy Holly's band, the Crickets. The group had cycled through several other names, including Johnny and the Moondogs, Japage 3, and Los Paranoias, before settling on the new identity.
George Harrison first saw the group perform on the 6th of February 1958 at Wilson Hall. He auditioned in March at Rory Storm's Morgue Skiffle Club, playing "Guitar Boogie Shuffle". Lennon initially resisted because Harrison was only 15, but after a second audition on the upper deck of a Liverpool bus, where Harrison played "Raunchy", Lennon allowed him to join as lead guitarist.
The surviving original members of the 1957 Quarrymen lineup reunited in 1997 for the 40th anniversary of the Woolton garden fete where Lennon and McCartney first met. Since 1998 they have performed internationally and released four studio albums: Open for Engagements (1994), Get Back - Together (1997), Songs We Remember (2004), and Grey Album (2012).