Cloud computing menas you are using someone elses computer and you are connecting over the internet to that comuter. Google's GMAIL and Microsofts HOTMAIL is cloud computing, you are using their email servers to keep look after your email. You don't need a mail server yourself, they provide one for you.
The great advantage is the savings made because you don't have to buy computer hardware. They manage all that for you. If you use Office 365 from Microsoft you have access to your files anywhere in the world and you can also work on them just by using a web browser.
Remember cloud computing is just computers in a big data centre that can be accessed from anywhere at anytime from a computer, laptop, tablet or phone.