I'm guessing this DVD was just burned by the school. If they didn't use a good brand of recordable media, it may not have good results elsewhere. Even with good brands, there's an element of your-mileage-may-vary when it comes to optical media. That brand of media may just not like both your laptop and desktop hard drives.
If you can ask the school to burn a disc for you, that would really be the best way. You can do what you want using what's called a video capture card. However, they're a bit expensive - the top seller in this category on Newegg is $70. (Link.)
I'd really look into if you can ask the school to make a copy for you instead. You can do what you want, yes, but it's not a cheap way to do it.