ignore helper monkey, he knows slightly more than you do, and its not all accurate.
Electric guitars use pick-ups which are transducers, that convert the vibrations from the strings into electric signals (which can be modified, w/ pedals), then converted back into audible sound through the amp.
Acoustic guitars use the hollowness of the body to amply the vibrations through the soundhole.
"Acoustic-electrics" (not the other way around), are acoustic guitars that have a build-in pre-amp (a little box to control the levels) which has a mic in it, which also you to plug it into an amp if you choose to do so.
If you by a acoustic-electric you shouldnt buy an amp. Its not like you're going to be doing any gigs or large performances. In actuality you should either buy a regular acoustic guitar, w/o a preamp. A acoustic-electric compared to a regular acoustic at the same price, which not be as good as the regular acoustic. Acoustic w/ a after-market pickup will sound better, and you can always mic the guitar. The only time you'll need headphones is if you have an electric guitar, even so I play my electric all through the nite w/ no amp. Unless its loud, you can play w/o an amp. Again if its for a class get an acoustic guitar w/o the electric components.
Good brands: (A nonshitty guitar is gonna be around $200, no less than $150. Remember guitar is a musical instrument not a toy, and instruments cost money. $200 is REALLY cheap when compared to other starting instruments)
Yamaha, Washburn, Alverez, Takamine, Seagull, Ibanez (I personally dont care too much for these, but not bad), Taylor (often expensive but on several occasions ive seen the Taylor Big Baby for $250)