Some of these answers give parts of the story, but not all of it.
Shortly after the lads signed with the BBC, its head of comedy, Michael Mills, began to refer to them as "the circus" in memos and whatnot, as no one quite seemed to know what they were doing; they'd signed to do a comedy series but even the lads themselves didn't really know what direction it would take. So, "Circus" was a given for the title.
The origin of "Flying" is less certain. Mills had occasionally referred to them as "Baron von Took's Flying Circus" for writer/comedian and BBC comedy advisor Barry Took, who was sort of the "scout" who got the BBC to sign them. But, other sources say the Pythons themselves added "Flying" because they liked the World War I sound of it (perhaps taking "Baron von Took" to its natural end, Baron von Richthofen) and not incidentally so viewers wouldn't initially think the show was an actual circus. Either way, "Flying Circus" was established.
Others here have covered the "Gwen Dibley" story, so I'll skip that. They wanted the name of the "operator" of the Flying Circus to suggest a show-business agent with the ethics of an albatross salesman. Cleese came up with "Python" because it sounded slithery and sinister and Idle thought of "Monty" for WWII Field Marshal Bernard "Monty" Montgomery, the British Army's version of U.S. Gen. George S. Patton.
Other titles suggested were "A Horse, a Bucket and a Spoon," "Bun, Wackett, Buzzard, Stubble and Boot," "Whither Canada?," "Owl-Stretching Time," "The Toad Elevating Moment" and "Ethel the Frog." The latter four were used in the series as episode or sketch titles.
As for the reason the group itself was merely "Monty Python," "Monty Python's Flying Circus" was the title of the TV series. The group itself never had an official name; "Monty Python" just naturally evolved. (In the fourth and final series, though, after the departure of John Cleese, the show was officially called "Monty Python.")
I hope this muddles things up for you. Bye for now. Keep your teeth clean. :)