Yes - you can play an electro-acoustic guitar without plugging it in!
Electro-acoustics will sound almost exactly like a normal acoustic guitar. The only difference being, if you wish to amp up your electro-acoustic you only need to plug it in rather than spend time using microphones to amplify it. The beauty of these guitars is that you still get the sound of an acoustic guitar but with the ability to amplify them to a much higher volume.
A semi-acoustic guitar is different; they are electric guitars but with addition hollow sound chambers built in. These guitars will not sound as good when they are not amplified.
Amplification
Conventional electric guitar pickups do not work on electro-acoustics because they cannot pickup the vibrations of non-magnetic materials. Instead, electro-acoustics possess microphones, transducers or piezoelectric pickups to capture the sound. Most electro-acoustics will also come fitted with a preamplifier which boosts the signal before it reaches the amplifier. Preamplifiers will come with tone controls so you can have a greater degree of control of your sound.
Origins
The first electro-acoustic was developed in 1966 by Charles H. Kaman. It was called the Ovation Round-back; was built from fibreglass, and was the first acoustic guitar which could be plugged in to an amp. Kaman's company, Kaman Music Corporation would subsequently become one of the largest musical instrument manufactures in the United States.
Hope that helps!