The most basic use case is that I use it as an online repository to access my files. If I have certain files that I need to access from multiple places, I upload them to Box.
The other use case is as a sharing mechanism. When I have large files that cannot be attached to an email, I upload the attachment to Box, and then I just share the link. I use it as a sharing mechanism.
The next use case is collaboration. For example, if I have an Excel spreadsheet that I'm updating and I want someone else to update it as well, I share the link with that person so that he can update the sheet on his end.
The beauty of Box is that you don't need to download the document on your computer to edit it and then upload it again. You can edit the content while the document is in the cloud.