My boyfriend just bought a brand new bike and was trying to teach me to ride it, and (I, being a complete noob) when I was standing with it I messed with the gears using the handle at a standstill. Changed the rear one from 3 to 1. He then started riding it and the the plastic part that helps it shift (I think) broke. You could hear a sudden crunching when he started riding it and the whole back part just snapped and broke. I assume it was that I just moved that, but was it? And why?
I would say, the gears weren't adjusted well, and chain drop out and brake the plastic that actually not very important. So it isn't your fault.
The real damage caused by changing gears while standing still is rub the cogwheels. So don't do it again.
The changing of gears itself is unlikely to cause any damage, you're just going to move the derailleurs back and forth until the chain prevents them moving further. If you really forced it you might break a shift lever or something. However, bad stuff is likely to happen if you heavily load the drivetrain while in the wrong gear. I.e. if you change gears while standing still then forcefully push down on the pedals when getting started, since the chain isn't on the right cog. There's a good chance then of the chain falling off or of it getting jammed somewhere, either leading to you falling off the bike, or potentially to some damage to the bike. With regards to your missing plastic bike: I don't know exactly what it was, but if the bike still works fine I wouldn't worry about it.
Also, your boyfriend. not you, is the fool in this situation. Changing gears while stationary isn't that crazy. Taking the bike and cranking heavily in the wrong gear is.
If you find yourself stationary on the wrong cog you just need to lift up the rear wheel and turn the crank forward a full revolution or two: that should get it into the right gear. With a little practice you can do this standing over the bike with your foot on the right pedal: lift the rear wheel off the ground slightly using your right hand on the saddle and push the crank with your right foot. Handy if you find yourself in the wrong gear at some traffic lights.
Unless the bike was very cheap, there's nothing that would break from that. The crunch of the chain on the rear gears and even the chain coming off are possible. You can just pop the chain back on the gears in the back. Over time this can cause issues, but unless you ride a lot and do this frequently, it would be a very long time. Mountain bikes especially are meant to be tough. If his brand new bike's gears broke he ought to be headed back to that shop for a refund.
And in the meantime, don't stop from messing! Get on that bike and ride around, mess around with it! If he did buy a very expensive new bike ask to ride his old one, or get yourself one for $50 on craigslist. You'll learn fastest by doing, and you'll make some mistakes, but the bike will forgive you. If you do something that makes a lot of noise, don't do it again :). Just keep yourself safe and have fun!