They originally formed with John Lennon, Paul McCartney, George Harrison and Stuart Sutcliffe (the drummer from 1960-61). Eventually Pete Best (drummer: 1960-62) joined them too. Sutcliffe is often credited, along with Lennon, as being the original inventor of The Beatles name. Sutcliffe left when he realized he had more talent as an artist than a musician, but to this day, many call him "The Fifth Beatle" but that title is debatable in itself since it was given to many former Beatles members, including Pete Best and even long time manager, Brian Epstein. Anyways, after leaving, Pete continue playing as their drummer. Eventually, Epstein, Lennon, McCartney and Harrison decided to dismiss Pete Best from the group (because of poor drumming) after their first recording session in Abbey Road Studios.
They decided to replace him with Ringo Starr, who left Rory Storm and the Hurricanes to go join them. And so the iconic faces of The Beatles were born! Shortly sometime after their recordings together, they began touring with much success. His drumming skills were regarded as the greatest in the world by Paul McCartney and it showed through the success of their tours and albums. And so that is the story of the Beatle's drummers in specific, but to-the-point detail.
Special Thanks to Wikipedia for providing me all of this. Much of this paragraph was paraphrased and not copied