I realize there have been many versions over the years. What's the story behind the design of the first Starship Enterprise?
The original Enterprise design and models were created by the art director Matt Jeffries. There is some small debate about the various sources of inspiration that he drew from, but in general he drew on his background in aviation to come up with a design that had the feel of an aircraft, but yet didn't violate any of Gene Roddenberry's rules (i.e. shouldn't look like a rocket, no fins, no wings, no smoke trails, crew of 150, fantastic speed, etc).
He wanted a design that felt futuristic, yet practical. After a bunch of attempts, they finally settled on the design that eventually became the Enterprise. It featured a saucer section that served as the crew quarters and a main hull that served as the primary work areas, with two enormous power plants extending from the main hull.