You can think of "the cloud" as being like an external hard drive that lives on the web somewhere. You can set up a drive on your computer that saves files to your cloud storage, but the process for you is the same as saving something on the desktop. You can also generally access your files from anywhere you have internet access, from any kind of device (like a laptop, tablet or phone).
Having an online copy of a document means it is safe from the elements, whether that means a glass of water teetering on your desk or a hole in the pocket where you keep your flash drive. In addition to having an extra backup copy, storing documents on the cloud allows you to continuously work on one document from multiple locations (home, office, train, wherever) without having to store multiple copies. It can also facilitate collaboration with others without having to email a document back and forth.
These are just a few of the many, many services available for cloud storage. This Wikipedia page lists even more options along with some information about each.
Of all of the services above, Penn+Box is the only one that is approved for storing sensitive Penn data, including non-confidential University data, FERPA data, and human subject research data (with IRB approval). Details on the ISC Box Data Sensitivity page.
For a nice roundup of information about privacy concerns on the other services, see this blog post from The Verge.