The man usually called the father of the modern circus never named his show a circus. A rival did, borrowing a word that had sat in the history books for over a thousand years.
The word is Latin circus, "ring, circular line." It is either taken from Greek kirkos, "a circle, a ring," or related to it. The Romans used it for their oval racing tracks, above all the Circus Maximus, a chariot course between the Palatine and Aventine hills. Its final stone version was about 400 meters long and could seat around 250,000 people.
English picked up the word in the late 1300s, but only to describe those Roman arenas. For centuries it was a classroom word.
The modern show began with Philip Astley, a horseman whom the V&A says is often credited as the father of the modern circus. In 1768 he and his wife Patty set up a riding school in London. Unlike his rivals, he rode in a circle rather than a straight line. The V&A gives 42 feet as the ideal ring diameter for this kind of riding. But Astley called his space a circle, and his building an amphitheatre.
The name came from competitors. On 4 November 1782, the songwriter Charles Dibdin and the horseman Charles Hughes opened the Royal Circus in London. Wikipedia credits Dibdin with coining the term. The Oxford English Dictionary's first citation for the modern sense is a 1791 book, The History of the Royal Circus, and Etymonline also dates that sense to 1791.
From there the word spread fast. It moved from the arena to the company and to the show itself. Etymonline records "traveling circus" from 1838. By 1869 a circus could mean any chaotic uproar.
The ring idea also gave English a separate, quieter sense. From the early 1700s, a circus could be buildings arranged in a ring, or a circular road. That is why London has Piccadilly Circus and Oxford Circus, which have nothing to do with clowns.
The Latin word has a small family of its own. Its diminutive, circulus, "small ring," passed through French to become circle.