I wouldn't bother with trying to build a case for an internal drive, the cost difference isn't that much and since you admit you don't know squat about this that wouldn't be the route to go.
I would check the daily deals (usually posted only Mon-Fri on some of the non-vendor blogs) posted as Consumerist.com ("Morning Deals"), Lifehacker.com, techbargins.com, newegg.com, amazon etc. They aggregate the best deals going around each day (or in the case of amazon and newegg only show their own sites deals) and there are literally external hard drive deals pretty much every day. You can get 1TB for under $100 (I recently got an Iomega 1.5TB external for $119 shipped) if you keep an eye out for the deals.

I wouldn't worry too much about what brand to get, as long as you've heard of the brand name (HP, Seagate, Western Digital, Iomega, Hitachi, etc) you should be fine, especially the the 500-1000GB drive size you're looking for; all of the kinks in manufacturing have been worked out as far as I know.

Don't waste your time with ebay or craigslist, you're rolling the dice with those. I wouldn't go to best buy over my dead body, you definitely want to buy online because you'll get a better deal due to warehouses not having the same overhead costs as regular stores.

USB drives are compatible with anything that has a USB jack - the only reason they may call it a desktop drive would be if it was bulkier than maybe a drive made extra thin (at additional cost) to make it easier to carry in your laptop bag. I'm not 100% sure but "laptop" drives may not have to plug into the wall for power like most external drives do.

Cache size won't matter to you since you're just using it for storage. If you're planning on running games off there or high-def movies to export to your TV then maybe I'd look for a bigger cache size but 32MB should do fine for you.

edit: make sure you have USB2.0, if you run an external drive off 1.0 then I hope you are a very patient person because it will take eons to transfer stuff. If your laptop was built in the last couple years you should be good to go. Follow the directions here to see if you have USB 2.0 if you don't already know: http://ask-leo.com/how_can_i_tell_if_i_have_usb_20.html