Electro-acoustics blur the line in the guitar world. You get the warm tone that comes with a nice acoustic, as well as the simple "plug in and play" option for live performance. Recording an electroacoustic guitar is a different matter, since it produces a lot of options and it is difficult to know which is best. As with all things in music, how you proceed is very much down to your personal taste, but there are two main options.
Connect your guitar into your amplifier using a guitar input cable. Recording your electroacoustic through an amp is essentially the same as recording an electric guitar, the only issue being that you will lose some of the warm acoustic tone if you use this method.
Experiment with microphone positions. Typically, your microphone should be about 6 inches from the amplifier's grille, pointing toward the centre of the speaker. This will give you a standard sound. Record a test run and see if you like the sound. Moving the microphone closer to the amp will give more definition, and moving it farther away will allow the sound to resonate more before being picked up by the microphone. Generally speaking, to get an "acoustic" sound, the microphone should be a little farther away; 1 or 2 feet from the amp will do the job.
Record your guitar part. If you feel like the notes aren't clearly defined enough on the recording, move the microphone closer to the amp.
Set your microphone up about 6 inches away from your guitar. If you're using one microphone, it's best to have it pointing toward the hole in the body of the guitar. This allows the microphone to pick up the best mix of sound from your guitar. The low-end notes resonate more toward the bridge, and the high-end notes are clearer around the neck.
Record a test and move the microphone as desired. Again, closer provides more definition and further away gives a more ambient, resonated sound. The hollow body will provide warmth and resonance anyway, so it shouldn't be necessary to have the microphone very far away.
Test both low and high notes to make sure both sound out to your liking. If you are fortunate enough to have two microphones, you can get a great sound by pointing one at the bridge and one at the 12th fret. This gives you a high-quality stereo recording, as one microphone picks up mainly high-end sounds, and the other gets the low sounds. This also means you can adjust the volume of the high-end and low-end sounds individually on your mixer/computer. This method is generally seen as the best option for getting an authentic "acoustic" sound.
Alternatively, you can combine the two methods if you have two microphones. This can give both the definition from the amp and the authenticity and warmth direct from your guitar. When choosing microphones to record your electroacoustic, it is best to go for a condenser microphone. Dynamic microphones are better suited to extreme noise and rock-star wear and tear. Condensers are more sensitive to high volume, but give a much better depth of sound for acoustic guitars. You can get either a large or small diaphragm microphone, but small ones are more adaptable, and can pick up a wider frequency of sound than the large diaphragms.