What is it? 
"Your files, anywhere: Any file you save to Dropbox also instantly saves to your computers, phones, and the Dropbox website.
Simple sharingShared folders allow people to work together on the same projects and documents."

Video tour (2:10)
Extensions and Apps

Note that many mobile applications integrate with Dropbox for file storage and retrieval.

Note that you can always access your files on any computer by logging into https://www.dropbox.com

Ideas for Education / Relationship to Other Tools
From Chronicle of Higher Ed: Receiving Files with Dropbox

Dropbox is a bit unique from the other tools presented in this workshop in that it is primarily a way to store and retrieve your own files "in the cloud," rather than starting from existing web-based content. But Dropbox is an incredibly useful productivity tool on its own, and it also works with many other web services and mobile applications to enhance storing, receiving, retrieving, and sharing all kinds of information.

Additional Ideas/Resources

Top 10 Clever Ideas for Using Dropbox - on the advanced side, but could be useful 

Review of numerous services similar to Dropbox, including Google Drive and iCloud