Dropbox is one of those services (free) I use all the time. I have my important files backed up to Dropbox and synced between my home computer and school computer. I can also access these files from any web browser and my smartphone and tablet. I was using a service (Drop.io) in the past to have students send files to my Dropbox, but it shut down. Now there is a new one.
FileStork is a free service that allows you to request files from people right into your Dropbox. This is a great way to collect work from students or files from colleagues.
It is easy to use. You simply send a request to someone via email, or set up a standing link that users can use. They click on the link (emailed or standing) and then upload their files. The files are then available to you in your Dropbox. You can even limit what kinds of files you will allow them to upload.