As I student, I've always worried (for good reason) that I'll forget my assignments at home, and receive a failing grade on an assignment that I worked on and completed.....except I left it at home. So the two things that have saved my butt during school are Dropbox and Google Docs.
You've most likely heard of Dropbox by now, but if you haven't it's a service that (a) synchronizes your files across multiple computers, as many computers as you like (I believe), and (b) backs up those same files to your Dropbox account on the web. I've used Dropbox for several years now on and off, but school is really when it comes in handy. I currently have my entire "School" linked via junction into my My Dropbox folder so I have access to all of my school files, even those from previous semesters.
The cool part about Dropbox is that it not only serves as a backup, it also helps when doing the assignments, especially if you regularly use more than one computer. For example, I have a netbook that I take to classes and a desktop at home. I can work on my homework at home, then leave to my day of classes and in between classes, I start up my netbook, and boom: there's my homework. I can work on it all I want and save the changes. Then when I get home, I can switch to my desktop, and boom: it's synced with the new work I did while I was gone.
To put it simply, Dropbox serves as a good way to carry homework and assignments around without ever having to worry about having a flashdrive, or having multiple versions of a file and wondering which one is the latest you've worked on. It also backs up all your data in case your computer crashes/is stolen/breaks.
The last best part about Dropbox is that you can also access your school assignments from online too. So if my English professor tells the class to bring in our essay drafts but I forget my flash drive while I'm walking out the door, all I have to do is go to my online Dropbox and download it on a school computer.
The only thing that I've got to say about Dropbox is this: always remember to close your files. I was working on an essay a few weeks ago and I thought, "Ok, I'll work on it in the morning, leave to class, then work on it in the break." It was a fine idea, except that I left my essay running in OpenOffice on my desktop, and programs like OpenOffice (and most likely Microsoft Word) put a lock on open files, so Dropbox can't sync them. So that taught me to be on the safe side and always close my assignments, or at least make sure they are synced.
(Dropbox also has an iPhone/Pod/Pad app in the App Store. A feature worth mentioning is the ability to share a link to any file via e-mail. So in a pinch, you could send a link of a vital assignment to a professor right from your phone/pod/pad.)
You've probably heard of Google Docs, but if you haven't, it's a wonderful online office suite offered by the one and only, Google. It has pretty much an entire office suite, most notably being a word processor, a spreadsheet creator, and a presentation creator.
The advantage of Google Docs to Dropbox is that there is no need to sync: it's always on the web. Instead of having to download your document to work on it, Google Docs lets you edit it right in your browser, or even print or create a PDF.
So the advantage is that you can edit it anywhere: no need to download on a school computer. The disadvantage to Dropbox is that it doesn't backup your files and it's available only online. Dropbox is nice because you could go someplace that doesn't have wifi, work on an assignment, then go to wifi and backup+sync. Google Docs is only in one place, which requires internet.
It's your choice, really. It kind of depends on the situation. I highly suggest both, or at least one.