How I Use It
I use Dropbox as my personal remote server. It is a great way to have remote access to files. In addition, Dropbox allows you to generate links to folders and files. This allows me to give links to students and parents (or post links on my blog) when files need to be downloaded. I don't need to email files anymore, if they are in my Dropbox I can share them easily.
In addition, I use another service called dropitto.me which is a free website that links to your Dropbox. Once you have an account a unique URL is generated for you (ie: dropitto.me/mynamehere). You can password protect your URL if you'd like. Once you give your URL, and site password, out, people can upload files directly to your Dropbox. I use DropItTo.me extensively as a way for students to turn in work from home and as a daily way for students/parents/colleagues to share files with my that are too big to email and don't play nice with other services like Google Drive.
I think Dropbox is key for creating your own server to keep documents and resources available to students even when they ar not in the building. Using a companion application like DropItTo.me is a great way to extend the capabilities of Dropbox, making more of a two-way street.