Dropbox: This is my number one web resource for anyone let alone students and teachers. If you use public computers, more than one computer, smart phones or just simply want to ensure everything is backed up safe and secure then Dropbox is your best friend. Change a file on one device and it's updated within seconds on all the others.
Even if you are offline - Dropbox will work faultlessly then synchronize your life the second you reconnect to the web.
Essentially, if you use dropbox you don't need to carry a USB key or computer to be on top of your digital life.
Google Docs: If you do not have access to Microsoft Office which is still the best productivity suite available hands down then Google Doc's is a very robust and free alternative. Google Doc's will allow you to store spreadsheets and documents online and edit them directly from the browser. Students can share their work and collaborate with peers in real time with Google Doc's.
Gmail: Gmail is the keys to the internet kingdom, The best calendar, email and IM and chat solution on the web. Totally free and will work seamlessly with your phone, laptop, tablet with a single hiccup.
FreeMind: Great mind mapping program, useful for brainstorming, outlining projects, and keeping notes.
Evernote: If you are a student who attends loads of different classes and has plenty of things going on in your life then Evernote will keep all your notes, audio recordings and photos in a highly logical and organised manner, backed up to the cloud and will talk with any of your smart devices. Kind of like Dropbox for note taking.