Strictly speaking, all guitars that do not rely on amplification to produce sound are 'acoustic'. It is only in the last fifty years though that the term has become necessary, since the invention of the electric guitar. Generally speaking, 'acoustic' implies steel-strung as distinct from nylon-strung instruments, which themselves are commonly known as 'classical' or 'Spanish' guitars.
In fact, the steel-strung guitar is a relatively recent development. The Nineteenth and early Twentieth Centuries saw an explosion of distinctively American folk styles, that today would be described loosely as 'hillbilly' music. The popularity of loud instruments such as banjos and fiddles meant that the Spanish guitar, with its gut strings, struggled to be heard.
The steel-strung guitar was created as a response to this problem. But steel strings exert a greater pull on the body and neck, and the features common to most modern acoustic and electric guitars were in turn developed in response to this issue by the two leading makers of the time - Martin and Gibson.
Classical guitars are reinforced by 'transverse bracing' - pieces of wood glued across the underside of the soundboard or 'top'. This provides enough strength to match the tension of nylon or gut strings, but a better solution was required for steel strings. A thicker top would of course hold up to higher tension, but would also result in a loss of volume. Martin's system of intersecting diagonal braces, known as the 'X-brace' system, solved this problem while retaining a relatively thin, resonant top.
The neck also required reinforcement - classical guitar necks are usually made of a single piece of wood, but this is not sufficient to withstand the tension of steel strings, and even very slight neck warping is enough to ruin the playability of an instrument. The resolution of this problem arrived in the form of a steel rod placed in a channel under the fretboard. Almost all acoustic and electric steel-strung guitars and basses made in the past century employ this device, which is known as a 'truss rod' - some even have two.
While Martin were essentially responsible for the design of the modern flat-top acoustic guitar, Gibson borrowed from violin makers to create the 'archtop' acoustic guitar, which has had a significant influence on the development of large-bodied 'semi-acoustic' electric guitars, popular with jazz players to this day.