An acoustic guitar that can be plugged into an amplifier and played like an electric is important to performers who want the sound of an unamplified instrument but need to be heard about the roar or clinking cocktail glasses and crowds. Oh sure, an acoustic guitar can be miked, but that means musicians must stay put, which is not always possible when the music you're playing makes you want to get up and move.
The idea of miking an acoustic guitar came about shortly after Georg Neumann invented the condenser microphone in 1928. On guitars, microphones are known as pickups, which are mounted over soundholes—not to capture the sound on its way out, but to electronically hijack it on its way in. For many decades, these sorts of after-market pickups were commonly added to acoustic guitars, but in recent years the pickups on acoustic-electric guitars have been integrated more elegantly and effectively into the instrument's design.
Of the great acoustic-guitar makers, Martin, Taylor and Breedlove have all made extraordinary acoustic-electrics, while manufacturers such as Fender and Gretsch, which are arguably better known for their electric guitars, have introduced amplified acoustics to round out their product catalogs. And while some people often lump hollow-body electric guitars in with acoustic-electrics, these instruments should really be considered electric guitars because their sound suffers without amplification. A true acoustic-electric should sound good whether it's plugged in or not.